On Monday, 29 August 2022 at 13:23:29 UTC+2, richar...@gmail.com wrote:
> On 8/29/22 3:30 AM, Skep Dick wrote:
> > On Monday, 29 August 2022 at 04:17:25 UTC+2, Ben Bacarisse wrote:
> >> So you do know what I meant. I thought so. Why did you pretend
> >> otherwise?
> > No, actually! I still have absolutely NO fucking clue what you mean by Σ(x=0, ∞)f(x) either! And for the exact same reason!
> >
> > Why is "∞" in the domain of Σ when x ∈ ℝ !?!?!
> >
> > But you can keep on pretending that you care about "clarity"; and you can keep on playing victim to "intentional misunderstanding".
> Because they are different parameters, and each parameter might have a
> different domain.
>
> Lots of things have different domains for their different parameters.

Shut up, obscurantist!

I am asking about the DISCONTINUITY between domains when the domains are different.

Number on top and number on bottom of Σ - no problem. Closed interval.
Number on bottom, infinity on top - problem. Open interval.

So the exact same nonsense you were spewing emerges again. If the domain of the parameter at bottom of Σ is N, and for all x in N : Finite(x) where is the jump/discontinuity at which Σ becomes an “infinite” series?

On Tuesday, 30 August 2022 at 12:36:13 UTC+2, Andy Walker wrote:
> On 30/08/2022 01:05, Ben Bacarisse wrote:
> > Skep Dick <skepd...@gmail.com> writes:
> >> So the fact that every student of yours abandons the epsilon-delta
> >> method the moment they cross ocer into a Physics lecture was no signal
> >> of importance to you?
> > If they did (and I have no idea if they did) why would I care?
> Probably unlike Ben, I spent a significant part of my career
> teaching maths to physicists. I never tried to teach them delta-
> epsilon analysis. In the UK, basic calculus is done at school, so
> they came to university with a decent basic knowledge already of
> how to differentiate and integrate. It was more important to teach
> them the more advanced things they needed /for physics/. Even more
> important to teach them the limits of what they knew -- "If /this/
> happens, you need to consult a mathematician." Much the same with
> engineers, and others who needed post-school mathematics.
> >> The fact that Physicists (to this day) use calculus with
> >> infinitesimals is of no importance to you?
> Not in my modules they didn't.
> > I'm interested in infinitesimals. I like infinitesimals. I wish the
> > 0.999... = 1 deniers knew how to use them.
> Likewise. But they're not a good way to do calculus. You
> either run into all the problems that Newton, Euler, ... ran into in
> the 18thC and which led to rigorous analysis in the 19thC, or you
> have to work with hyperreals or surreals or similar, and that's an
> awful lot of extra baggage to carry around for something so simple
> [for most physics/engineering purposes]. It's more fruitful to
> teach limits informally, and emphasise that this is for "well-
> behaved functions" *only*. Good and bad behaviour illustrated by
> examples rather than by formal definitions.
>
> --
> Andy Walker, Nottingham.
> Andy's music pages: www.cuboid.me.uk/andy/Music
> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Handel

Here is the crux… if infinitesimals don’t exist then dy/dx is division by zero.

The rest is skirting around this gaping fuckup.

On 8/30/22 7:16 AM, Skep Dick wrote:
> On Tuesday, 30 August 2022 at 12:36:13 UTC+2, Andy Walker wrote:
>> On 30/08/2022 01:05, Ben Bacarisse wrote:
>>> Skep Dick <skepd...@gmail.com> writes:
>>>> So the fact that every student of yours abandons the epsilon-delta
>>>> method the moment they cross ocer into a Physics lecture was no signal
>>>> of importance to you?
>>> If they did (and I have no idea if they did) why would I care?
>> Probably unlike Ben, I spent a significant part of my career
>> teaching maths to physicists. I never tried to teach them delta-
>> epsilon analysis. In the UK, basic calculus is done at school, so
>> they came to university with a decent basic knowledge already of
>> how to differentiate and integrate. It was more important to teach
>> them the more advanced things they needed /for physics/. Even more
>> important to teach them the limits of what they knew -- "If /this/
>> happens, you need to consult a mathematician." Much the same with
>> engineers, and others who needed post-school mathematics.
>>>> The fact that Physicists (to this day) use calculus with
>>>> infinitesimals is of no importance to you?
>> Not in my modules they didn't.
>>> I'm interested in infinitesimals. I like infinitesimals. I wish the
>>> 0.999... = 1 deniers knew how to use them.
>> Likewise. But they're not a good way to do calculus. You
>> either run into all the problems that Newton, Euler, ... ran into in
>> the 18thC and which led to rigorous analysis in the 19thC, or you
>> have to work with hyperreals or surreals or similar, and that's an
>> awful lot of extra baggage to carry around for something so simple
>> [for most physics/engineering purposes]. It's more fruitful to
>> teach limits informally, and emphasise that this is for "well-
>> behaved functions" *only*. Good and bad behaviour illustrated by
>> examples rather than by formal definitions.
>>
>> --
>> Andy Walker, Nottingham.
>> Andy's music pages: www.cuboid.me.uk/andy/Music
>> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Handel
>
> Here is the crux… if infinitesimals don’t exist then dy/dx is division by zero.
>
> The rest is skirting around this gaping fuckup.

No, because in Calculus, we are DEFINING a form of infinitesimals and
the math that works with them as an extension of the Real Numbers. This
definition is based on limit theory and it all works together.

On 8/30/22 7:14 AM, Skep Dick wrote:
> On Monday, 29 August 2022 at 13:23:29 UTC+2, richar...@gmail.com wrote:
>> On 8/29/22 3:30 AM, Skep Dick wrote:
>>> On Monday, 29 August 2022 at 04:17:25 UTC+2, Ben Bacarisse wrote:
>>>> So you do know what I meant. I thought so. Why did you pretend
>>>> otherwise?
>>> No, actually! I still have absolutely NO fucking clue what you mean by Σ(x=0, ∞)f(x) either! And for the exact same reason!
>>>
>>> Why is "∞" in the domain of Σ when x ∈ ℝ !?!?!
>>>
>>> But you can keep on pretending that you care about "clarity"; and you can keep on playing victim to "intentional misunderstanding".
>> Because they are different parameters, and each parameter might have a
>> different domain.
>>
>> Lots of things have different domains for their different parameters.
>
> Shut up, obscurantist!
>
> I am asking about the DISCONTINUITY between domains when the domains are different.
>
> Number on top and number on bottom of Σ - no problem. Closed interval.
> Number on bottom, infinity on top - problem. Open interval.
>
> So the exact same nonsense you were spewing emerges again. If the domain of the parameter at bottom of Σ is N, and for all x in N : Finite(x) where is the jump/discontinuity at which Σ becomes an “infinite” series?
>

Why is that a problem. The operation is just overloaded based on two
types of parameters. When infinity gets used, we just activate limit
theory to evaluate.

Thus Sum to infinity becomes the limit of Sum to N as N goes to infinity.

On Tuesday, 30 August 2022 at 12:36:13 UTC+2, Andy Walker wrote:
> On 30/08/2022 01:05, Ben Bacarisse wrote:
> > Skep Dick <skepd...@gmail.com> writes:
> >> So the fact that every student of yours abandons the epsilon-delta
> >> method the moment they cross ocer into a Physics lecture was no signal
> >> of importance to you?
> > If they did (and I have no idea if they did) why would I care?
> Probably unlike Ben, I spent a significant part of my career
> teaching maths to physicists. I never tried to teach them delta-
> epsilon analysis. In the UK, basic calculus is done at school, so
> they came to university with a decent basic knowledge already of
> how to differentiate and integrate. It was more important to teach
> them the more advanced things they needed /for physics/. Even more
> important to teach them the limits of what they knew -- "If /this/
> happens, you need to consult a mathematician." Much the same with
> engineers, and others who needed post-school mathematics.
> >> The fact that Physicists (to this day) use calculus with
> >> infinitesimals is of no importance to you?
> Not in my modules they didn't.
> > I'm interested in infinitesimals. I like infinitesimals. I wish the
> > 0.999... = 1 deniers knew how to use them.
> Likewise. But they're not a good way to do calculus. You
> either run into all the problems that Newton, Euler, ... ran into in
> the 18thC and which led to rigorous analysis in the 19thC, or you
> have to work with hyperreals or surreals or similar, and that's an
> awful lot of extra baggage to carry around for something so simple
> [for most physics/engineering purposes]. It's more fruitful to
> teach limits informally, and emphasise that this is for "well-
> behaved functions" *only*. Good and bad behaviour illustrated by
> examples rather than by formal definitions.
>
> --
> Andy Walker, Nottingham.
> Andy's music pages: www.cuboid.me.uk/andy/Music
> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Handel

Take the usual way for calculating the derivative of f(x)=x^2

f’(x) =(f(x +dx) - f(x)) / dx = …… = 2x + dx

At this point we say stuff like “dx is so small we can basically ignore it” and conclude f’(x)=2x

But who really buys this argument? If dx is so small that we can ignore it then ignore it sooner…

f’(x) =(f(x +dx) - f(x)) / dx = (f(x) - f(x))/dx = 0

On Tuesday, 30 August 2022 at 13:38:35 UTC+2, richar...@gmail.com wrote:
> On 8/30/22 7:14 AM, Skep Dick wrote:
> > On Monday, 29 August 2022 at 13:23:29 UTC+2, richar...@gmail.com wrote:
> >> On 8/29/22 3:30 AM, Skep Dick wrote:
> >>> On Monday, 29 August 2022 at 04:17:25 UTC+2, Ben Bacarisse wrote:
> >>>> So you do know what I meant. I thought so. Why did you pretend
> >>>> otherwise?
> >>> No, actually! I still have absolutely NO fucking clue what you mean by Σ(x=0, ∞)f(x) either! And for the exact same reason!
> >>>
> >>> Why is "∞" in the domain of Σ when x ∈ ℝ !?!?!
> >>>
> >>> But you can keep on pretending that you care about "clarity"; and you can keep on playing victim to "intentional misunderstanding".
> >> Because they are different parameters, and each parameter might have a
> >> different domain.
> >>
> >> Lots of things have different domains for their different parameters.
> >
> > Shut up, obscurantist!
> >
> > I am asking about the DISCONTINUITY between domains when the domains are different.
> >
> > Number on top and number on bottom of Σ - no problem. Closed interval.
> > Number on bottom, infinity on top - problem. Open interval.
> >
> > So the exact same nonsense you were spewing emerges again. If the domain of the parameter at bottom of Σ is N, and for all x in N : Finite(x) where is the jump/discontinuity at which Σ becomes an “infinite” series?
> >
> Why is that a problem. The operation is just overloaded based on two
> types of parameters. When infinity gets used, we just activate limit
> theory to evaluate.
> Thus Sum to infinity becomes the limit of Sum to N as N goes to infinity.

Isn’t limit theory defined in terms of infinite sums?!?!?

Can you spell “circular dependency”?

On Tuesday, 30 August 2022 at 13:29:53 UTC+2, richar...@gmail.com wrote:
> On 8/30/22 7:16 AM, Skep Dick wrote:
> > On Tuesday, 30 August 2022 at 12:36:13 UTC+2, Andy Walker wrote:
> >> On 30/08/2022 01:05, Ben Bacarisse wrote:
> >>> Skep Dick <skepd...@gmail.com> writes:
> >>>> So the fact that every student of yours abandons the epsilon-delta
> >>>> method the moment they cross ocer into a Physics lecture was no signal
> >>>> of importance to you?
> >>> If they did (and I have no idea if they did) why would I care?
> >> Probably unlike Ben, I spent a significant part of my career
> >> teaching maths to physicists. I never tried to teach them delta-
> >> epsilon analysis. In the UK, basic calculus is done at school, so
> >> they came to university with a decent basic knowledge already of
> >> how to differentiate and integrate. It was more important to teach
> >> them the more advanced things they needed /for physics/. Even more
> >> important to teach them the limits of what they knew -- "If /this/
> >> happens, you need to consult a mathematician." Much the same with
> >> engineers, and others who needed post-school mathematics.
> >>>> The fact that Physicists (to this day) use calculus with
> >>>> infinitesimals is of no importance to you?
> >> Not in my modules they didn't.
> >>> I'm interested in infinitesimals. I like infinitesimals. I wish the
> >>> 0.999... = 1 deniers knew how to use them.
> >> Likewise. But they're not a good way to do calculus. You
> >> either run into all the problems that Newton, Euler, ... ran into in
> >> the 18thC and which led to rigorous analysis in the 19thC, or you
> >> have to work with hyperreals or surreals or similar, and that's an
> >> awful lot of extra baggage to carry around for something so simple
> >> [for most physics/engineering purposes]. It's more fruitful to
> >> teach limits informally, and emphasise that this is for "well-
> >> behaved functions" *only*. Good and bad behaviour illustrated by
> >> examples rather than by formal definitions.
> >>
> >> --
> >> Andy Walker, Nottingham.
> >> Andy's music pages: www.cuboid.me.uk/andy/Music
> >> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Handel
> >
> > Here is the crux… if infinitesimals don’t exist then dy/dx is division by zero.
> >
> > The rest is skirting around this gaping fuckup.
> No, because in Calculus, we are DEFINING a form of infinitesimals and
> the math that works with them as an extension of the Real Numbers. This
> definition is based on limit theory and it all works together.

Ooooh, so when you said you were workin in R you were lying?

On 8/30/22 7:42 AM, Skep Dick wrote:
> On Tuesday, 30 August 2022 at 13:29:53 UTC+2, richar...@gmail.com wrote:
>> On 8/30/22 7:16 AM, Skep Dick wrote:
>>> On Tuesday, 30 August 2022 at 12:36:13 UTC+2, Andy Walker wrote:
>>>> On 30/08/2022 01:05, Ben Bacarisse wrote:
>>>>> Skep Dick <skepd...@gmail.com> writes:
>>>>>> So the fact that every student of yours abandons the epsilon-delta
>>>>>> method the moment they cross ocer into a Physics lecture was no signal
>>>>>> of importance to you?
>>>>> If they did (and I have no idea if they did) why would I care?
>>>> Probably unlike Ben, I spent a significant part of my career
>>>> teaching maths to physicists. I never tried to teach them delta-
>>>> epsilon analysis. In the UK, basic calculus is done at school, so
>>>> they came to university with a decent basic knowledge already of
>>>> how to differentiate and integrate. It was more important to teach
>>>> them the more advanced things they needed /for physics/. Even more
>>>> important to teach them the limits of what they knew -- "If /this/
>>>> happens, you need to consult a mathematician." Much the same with
>>>> engineers, and others who needed post-school mathematics.
>>>>>> The fact that Physicists (to this day) use calculus with
>>>>>> infinitesimals is of no importance to you?
>>>> Not in my modules they didn't.
>>>>> I'm interested in infinitesimals. I like infinitesimals. I wish the
>>>>> 0.999... = 1 deniers knew how to use them.
>>>> Likewise. But they're not a good way to do calculus. You
>>>> either run into all the problems that Newton, Euler, ... ran into in
>>>> the 18thC and which led to rigorous analysis in the 19thC, or you
>>>> have to work with hyperreals or surreals or similar, and that's an
>>>> awful lot of extra baggage to carry around for something so simple
>>>> [for most physics/engineering purposes]. It's more fruitful to
>>>> teach limits informally, and emphasise that this is for "well-
>>>> behaved functions" *only*. Good and bad behaviour illustrated by
>>>> examples rather than by formal definitions.
>>>>
>>>> --
>>>> Andy Walker, Nottingham.
>>>> Andy's music pages: www.cuboid.me.uk/andy/Music
>>>> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Handel
>>>
>>> Here is the crux… if infinitesimals don’t exist then dy/dx is division by zero.
>>>
>>> The rest is skirting around this gaping fuckup.
>> No, because in Calculus, we are DEFINING a form of infinitesimals and
>> the math that works with them as an extension of the Real Numbers. This
>> definition is based on limit theory and it all works together.
>
> Ooooh, so when you said you were workin in R you were lying?

No, because when we do Calculus, the only spot the infinitesimals occurs
is inside the math to actually evaluate the Calculus operations.

We don't just have a "loose" dx, but it is always as part of an integral
or derivative operation, so dx itself isn't an infinitesimal, but the
"d" "operator" that needs a matching "d" for a derivative or an integral
to give it full definition.

Its another thing that may give you notation a challenge, as "d" isn't
actually an operator itself, but part of a multi-symbol thingy that
defines an operation. There is no simple defintion for the "d" itself,
only what it means when paired with its partner.

On Tuesday, 30 August 2022 at 14:01:40 UTC+2, richar...@gmail.com wrote:
> On 8/30/22 7:42 AM, Skep Dick wrote:
> > On Tuesday, 30 August 2022 at 13:29:53 UTC+2, richar...@gmail.com wrote:
> >> On 8/30/22 7:16 AM, Skep Dick wrote:
> >>> On Tuesday, 30 August 2022 at 12:36:13 UTC+2, Andy Walker wrote:
> >>>> On 30/08/2022 01:05, Ben Bacarisse wrote:
> >>>>> Skep Dick <skepd...@gmail.com> writes:
> >>>>>> So the fact that every student of yours abandons the epsilon-delta
> >>>>>> method the moment they cross ocer into a Physics lecture was no signal
> >>>>>> of importance to you?
> >>>>> If they did (and I have no idea if they did) why would I care?
> >>>> Probably unlike Ben, I spent a significant part of my career
> >>>> teaching maths to physicists. I never tried to teach them delta-
> >>>> epsilon analysis. In the UK, basic calculus is done at school, so
> >>>> they came to university with a decent basic knowledge already of
> >>>> how to differentiate and integrate. It was more important to teach
> >>>> them the more advanced things they needed /for physics/. Even more
> >>>> important to teach them the limits of what they knew -- "If /this/
> >>>> happens, you need to consult a mathematician." Much the same with
> >>>> engineers, and others who needed post-school mathematics.
> >>>>>> The fact that Physicists (to this day) use calculus with
> >>>>>> infinitesimals is of no importance to you?
> >>>> Not in my modules they didn't.
> >>>>> I'm interested in infinitesimals. I like infinitesimals. I wish the
> >>>>> 0.999... = 1 deniers knew how to use them.
> >>>> Likewise. But they're not a good way to do calculus. You
> >>>> either run into all the problems that Newton, Euler, ... ran into in
> >>>> the 18thC and which led to rigorous analysis in the 19thC, or you
> >>>> have to work with hyperreals or surreals or similar, and that's an
> >>>> awful lot of extra baggage to carry around for something so simple
> >>>> [for most physics/engineering purposes]. It's more fruitful to
> >>>> teach limits informally, and emphasise that this is for "well-
> >>>> behaved functions" *only*. Good and bad behaviour illustrated by
> >>>> examples rather than by formal definitions.
> >>>>
> >>>> --
> >>>> Andy Walker, Nottingham.
> >>>> Andy's music pages: www.cuboid.me.uk/andy/Music
> >>>> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Handel
> >>>
> >>> Here is the crux… if infinitesimals don’t exist then dy/dx is division by zero.
> >>>
> >>> The rest is skirting around this gaping fuckup.
> >> No, because in Calculus, we are DEFINING a form of infinitesimals and
> >> the math that works with them as an extension of the Real Numbers. This
> >> definition is based on limit theory and it all works together.
> >
> > Ooooh, so when you said you were workin in R you were lying?
> No, because when we do Calculus, the only spot the infinitesimals occurs
> is inside the math to actually evaluate the Calculus operations.
>
> We don't just have a "loose" dx, but it is always as part of an integral
> or derivative operation, so dx itself isn't an infinitesimal, but the
> "d" "operator" that needs a matching "d" for a derivative or an integral
> to give it full definition.
>
> Its another thing that may give you notation a challenge, as "d" isn't
> actually an operator itself, but part of a multi-symbol thingy that
> defines an operation. There is no simple defintion for the "d" itself,
> only what it means when paired with its partner.

Do you want to keep digging?

I will get you a shovel.

On 8/30/22 7:40 AM, Skep Dick wrote:
> On Tuesday, 30 August 2022 at 13:38:35 UTC+2, richar...@gmail.com wrote:
>> On 8/30/22 7:14 AM, Skep Dick wrote:
>>> On Monday, 29 August 2022 at 13:23:29 UTC+2, richar...@gmail.com wrote:
>>>> On 8/29/22 3:30 AM, Skep Dick wrote:
>>>>> On Monday, 29 August 2022 at 04:17:25 UTC+2, Ben Bacarisse wrote:
>>>>>> So you do know what I meant. I thought so. Why did you pretend
>>>>>> otherwise?
>>>>> No, actually! I still have absolutely NO fucking clue what you mean by Σ(x=0, ∞)f(x) either! And for the exact same reason!
>>>>>
>>>>> Why is "∞" in the domain of Σ when x ∈ ℝ !?!?!
>>>>>
>>>>> But you can keep on pretending that you care about "clarity"; and you can keep on playing victim to "intentional misunderstanding".
>>>> Because they are different parameters, and each parameter might have a
>>>> different domain.
>>>>
>>>> Lots of things have different domains for their different parameters.
>>>
>>> Shut up, obscurantist!
>>>
>>> I am asking about the DISCONTINUITY between domains when the domains are different.
>>>
>>> Number on top and number on bottom of Σ - no problem. Closed interval.
>>> Number on bottom, infinity on top - problem. Open interval.
>>>
>>> So the exact same nonsense you were spewing emerges again. If the domain of the parameter at bottom of Σ is N, and for all x in N : Finite(x) where is the jump/discontinuity at which Σ becomes an “infinite” series?
>>>
>> Why is that a problem. The operation is just overloaded based on two
>> types of parameters. When infinity gets used, we just activate limit
>> theory to evaluate.
>> Thus Sum to infinity becomes the limit of Sum to N as N goes to infinity.
>
> Isn’t limit theory defined in terms of infinite sums?!?!?
>
> Can you spell “circular dependency”?

Nope, not at all. I gave you the definition previously and there wasn't
a sum in it.

Lim(x->∞) f(x) is defined to be L if and only if:

For all e > 0 there exist a number X such that for all x > X it is true
that the magnatude of f(x) - L is < e.

Not a sum in sight.

For the Sum F(x) is just the F(N) and is the FINITE sum from start to N,
so rather than x being a real, it is an integer.

On 8/30/22 8:04 AM, Skep Dick wrote:
> On Tuesday, 30 August 2022 at 14:01:40 UTC+2, richar...@gmail.com wrote:
>> On 8/30/22 7:42 AM, Skep Dick wrote:
>>> On Tuesday, 30 August 2022 at 13:29:53 UTC+2, richar...@gmail.com wrote:
>>>> On 8/30/22 7:16 AM, Skep Dick wrote:
>>>>> On Tuesday, 30 August 2022 at 12:36:13 UTC+2, Andy Walker wrote:
>>>>>> On 30/08/2022 01:05, Ben Bacarisse wrote:
>>>>>>> Skep Dick <skepd...@gmail.com> writes:
>>>>>>>> So the fact that every student of yours abandons the epsilon-delta
>>>>>>>> method the moment they cross ocer into a Physics lecture was no signal
>>>>>>>> of importance to you?
>>>>>>> If they did (and I have no idea if they did) why would I care?
>>>>>> Probably unlike Ben, I spent a significant part of my career
>>>>>> teaching maths to physicists. I never tried to teach them delta-
>>>>>> epsilon analysis. In the UK, basic calculus is done at school, so
>>>>>> they came to university with a decent basic knowledge already of
>>>>>> how to differentiate and integrate. It was more important to teach
>>>>>> them the more advanced things they needed /for physics/. Even more
>>>>>> important to teach them the limits of what they knew -- "If /this/
>>>>>> happens, you need to consult a mathematician." Much the same with
>>>>>> engineers, and others who needed post-school mathematics.
>>>>>>>> The fact that Physicists (to this day) use calculus with
>>>>>>>> infinitesimals is of no importance to you?
>>>>>> Not in my modules they didn't.
>>>>>>> I'm interested in infinitesimals. I like infinitesimals. I wish the
>>>>>>> 0.999... = 1 deniers knew how to use them.
>>>>>> Likewise. But they're not a good way to do calculus. You
>>>>>> either run into all the problems that Newton, Euler, ... ran into in
>>>>>> the 18thC and which led to rigorous analysis in the 19thC, or you
>>>>>> have to work with hyperreals or surreals or similar, and that's an
>>>>>> awful lot of extra baggage to carry around for something so simple
>>>>>> [for most physics/engineering purposes]. It's more fruitful to
>>>>>> teach limits informally, and emphasise that this is for "well-
>>>>>> behaved functions" *only*. Good and bad behaviour illustrated by
>>>>>> examples rather than by formal definitions.
>>>>>>
>>>>>> --
>>>>>> Andy Walker, Nottingham.
>>>>>> Andy's music pages: www.cuboid.me.uk/andy/Music
>>>>>> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Handel
>>>>>
>>>>> Here is the crux… if infinitesimals don’t exist then dy/dx is division by zero.
>>>>>
>>>>> The rest is skirting around this gaping fuckup.
>>>> No, because in Calculus, we are DEFINING a form of infinitesimals and
>>>> the math that works with them as an extension of the Real Numbers. This
>>>> definition is based on limit theory and it all works together.
>>>
>>> Ooooh, so when you said you were workin in R you were lying?
>> No, because when we do Calculus, the only spot the infinitesimals occurs
>> is inside the math to actually evaluate the Calculus operations.
>>
>> We don't just have a "loose" dx, but it is always as part of an integral
>> or derivative operation, so dx itself isn't an infinitesimal, but the
>> "d" "operator" that needs a matching "d" for a derivative or an integral
>> to give it full definition.
>>
>> Its another thing that may give you notation a challenge, as "d" isn't
>> actually an operator itself, but part of a multi-symbol thingy that
>> defines an operation. There is no simple defintion for the "d" itself,
>> only what it means when paired with its partner.
>
> Do you want to keep digging?
>
> I will get you a shovel.

Maybe you can use your notation to handle the "d" operator.

Don't know what other hole you are commenting about.

On 8/30/22 6:52 AM, Skep Dick wrote:
> On Tuesday, 30 August 2022 at 02:28:32 UTC+2, richar...@gmail.com wrote:
>> On 8/29/22 8:21 AM, Skep Dick wrote:
>>> On Monday, 29 August 2022 at 14:12:35 UTC+2, richar...@gmail.com wrote:
>>>> So, since they are not defined as parameterized, maybe you system can't
>>>> handle it.
>>> Shame! Are you tripping up over syntax?
>>>
>>> sigma(start, end, f)
>>>
>>> In [1]: def sigma(start, end, f):
>>> ...: sum = 0
>>> ...: for x in range(start, end + 1):
>>> ...: sum = sum + f(x)
>>> ...: return sum
>>>
>>> In [2]: sigma(0,2, (lambda x:x))
>>> Out[2]: 3
>>>
>>> In [3]: sigma(0,2, (lambda x: 2**x))
>>> Out[3]: 7
>> yes, but that program won't work with end = ∞, as it will never finish.
> I think you are lying. The program will work just fine, but it will never finish.
> It is just an infinite loop.
>
> The value of “sum” inside the program will continue to accumulate.

If you want ot call that "work", you never get an answer, not even an
approximat one.

>
>> And ∞ + 1 isn't a defined value. The problem being that if you try to
>> treat ∞ just like any other number, you get into all sorts of contradictins.
> That is just a straw man. I am not treating it as a number any more than you are treating it as a number when you put it on top of Σ.
>
> We can just treat the range() function as polymorphic. It can range over [0, end] for finite values, or [0, ∞) for infinite values.
>
>>>> So the problem is YOU are treating it as part of a continuum that it
>>>> isn't part of.
>>> What? Yes YOU are treating it as part of the continuum! Not me!
>>>
>>> [0, ∞) UNION ∞ is [0,∞]
>> except that [0, ∞) and ∞ are different sorts of things.
> Yes, and? Your UNION operator doesn’t seem to mind the type mismatch?

Union was an Operator on TYPES, creating the TYPE (which might not have
a name) that represents something that can be a Real or an Infinte.

>
> So what is the result of the operation UNiON([0, ∞), ∞) ?

UNiON([0, ∞), ∞), if you let it take that sort of thing.

I defined UNION on TYPES, not ranges, but it could be.

>
>>>
>> But MY system doesn't have a contiuum of [0, ∞]
> Then what do you mean by [0, ∞) UNION ∞ ?

The combination of the Set of Real Nubers from 0 to an unbounded
maximum, unioned the infinite value ∞. it doesn't need to be able to be
reduced.

>
>> or a parametric overload of the limit operatior, so it can't be my system.
> Ohhh! So you are using a different system to Ben? You guys should have a chat!
>
> He says lim() is overloaded for asymptotic and non-asymptotic limits.

Right, it just isn't a parameteric overload. non-asymptotic limits, i.e.
limits to a real value, have one definition, and asymptotic limits, i.e.
limits to an infinite value. have a different (but related) definition.

>
>> All that is an effet of you trying to redefine things into a different
>> fundamental framework.
> I am not redefining anything. I am UNIFYING your mess into a framework that deals with polymorphism.
>
> Generic types.

SO do so. If you can't, that is YOUR problem for not using a powerful
enough system.

That would be like signing up to the Indy 500 with a Go-Kart.

>
>> Any problem there-of, are YOUR problems, not mine.
> Your mess is everyone’s problem.
>

It is well defined. If you can't translate that to your system, then
your system isn't powerful enough.

>
>> I know how to take a limit, and how to define it.
> You don’t know shit. You can’t even define “define”, but you think you can define a limit.
>
>> The fact that you have
>> issues converting from a system based on limits to one where you are
>> bound by computability is your own problem, not mine.
> It is difficult to explain to an idiot that he doesn’t understand.
>
> I am no more bound by computability than you are bound by your “foundation”.
>
> I am merely demonstrating that my foundation is more foundational.

It seems your foundation doesn't HAVE a foundation to stand on.

On Tuesday, 30 August 2022 at 19:38:38 UTC+8, skepd...@gmail.com wrote:
> On Tuesday, 30 August 2022 at 12:36:13 UTC+2, Andy Walker wrote:
> > On 30/08/2022 01:05, Ben Bacarisse wrote:
> > > Skep Dick <skepd...@gmail.com> writes:
> > >> So the fact that every student of yours abandons the epsilon-delta
> > >> method the moment they cross ocer into a Physics lecture was no signal
> > >> of importance to you?
> > > If they did (and I have no idea if they did) why would I care?
> > Probably unlike Ben, I spent a significant part of my career
> > teaching maths to physicists. I never tried to teach them delta-
> > epsilon analysis. In the UK, basic calculus is done at school, so
> > they came to university with a decent basic knowledge already of
> > how to differentiate and integrate. It was more important to teach
> > them the more advanced things they needed /for physics/. Even more
> > important to teach them the limits of what they knew -- "If /this/
> > happens, you need to consult a mathematician." Much the same with
> > engineers, and others who needed post-school mathematics.
> > >> The fact that Physicists (to this day) use calculus with
> > >> infinitesimals is of no importance to you?
> > Not in my modules they didn't.
> > > I'm interested in infinitesimals. I like infinitesimals. I wish the
> > > 0.999... = 1 deniers knew how to use them.
> > Likewise. But they're not a good way to do calculus. You
> > either run into all the problems that Newton, Euler, ... ran into in
> > the 18thC and which led to rigorous analysis in the 19thC, or you
> > have to work with hyperreals or surreals or similar, and that's an
> > awful lot of extra baggage to carry around for something so simple
> > [for most physics/engineering purposes]. It's more fruitful to
> > teach limits informally, and emphasise that this is for "well-
> > behaved functions" *only*. Good and bad behaviour illustrated by
> > examples rather than by formal definitions.
> >
> > --
> > Andy Walker, Nottingham.
> > Andy's music pages: www.cuboid.me.uk/andy/Music
> > Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Handel
> Take the usual way for calculating the derivative of f(x)=x^2
>
> f’(x) =(f(x +dx) - f(x)) / dx = …… = 2x + dx
>
> At this point we say stuff like “dx is so small we can basically ignore it” and conclude f’(x)=2x
>
> But who really buys this argument? If dx is so small that we can ignore it then ignore it sooner…
>
> f’(x) =(f(x +dx) - f(x)) / dx = (f(x) - f(x))/dx = 0

I think I solved this problem.
The basic problem of calculus is the definition of slope of a tangent line.
slope= (f(x+h)-f(x))/h .... (h is ?)
The issue is that tangent lines require h be zero (otherwise, it is a secant line
defined by two points), which causes diving zero problems.

My idea is:
Let g(x,h)= (f(x+h)-f(x))/h

g'(x,h)≡ identical to g(x,h) and g'(x,0) is defined (h is defined in the set of infinitesimal).
Slope of tangent line is defined as g'(x,0). There can only be at most one version of g'.
[note] g' and g are looked very close (only different in one point), but the
essence are very different. g' is defined on a deleted set.

Result: Power functions have such g' (exact) slope function, therefore, usual
x^2,x^3,.. geometric analyses are exactly correct.
Exponential and trigonometric functions are questionable (but the error is infinitesimal).
And probably many concepts like 'continuous'... developed in the limit theory
and 'real analysis' can be saved. All begin from a definition of '∞'.

---
The use of "dy/dx" is different, probably some kind of logic. I said nothing about this.

On Tuesday, 30 August 2022 at 14:52:37 UTC+2, richar...@gmail.com wrote:
> On 8/30/22 6:52 AM, Skep Dick wrote:
> > On Tuesday, 30 August 2022 at 02:28:32 UTC+2, richar...@gmail.com wrote:
> >> On 8/29/22 8:21 AM, Skep Dick wrote:
> >>> On Monday, 29 August 2022 at 14:12:35 UTC+2, richar...@gmail.com wrote:
> >>>> So, since they are not defined as parameterized, maybe you system can't
> >>>> handle it.
> >>> Shame! Are you tripping up over syntax?
> >>>
> >>> sigma(start, end, f)
> >>>
> >>> In [1]: def sigma(start, end, f):
> >>> ...: sum = 0
> >>> ...: for x in range(start, end + 1):
> >>> ...: sum = sum + f(x)
> >>> ...: return sum
> >>>
> >>> In [2]: sigma(0,2, (lambda x:x))
> >>> Out[2]: 3
> >>>
> >>> In [3]: sigma(0,2, (lambda x: 2**x))
> >>> Out[3]: 7
> >> yes, but that program won't work with end = ∞, as it will never finish.
> > I think you are lying. The program will work just fine, but it will never finish.
> > It is just an infinite loop.
> >
> > The value of “sum” inside the program will continue to accumulate.
> If you want ot call that "work", you never get an answer, not even an
> approximat one.

Of course I call it work. You don’t? An approximate answer is incomplete work!
At ANY point of the execution you can sample the value of “sum”. So you can always know the value after n

Surely you of all people would understand? You are giving Olcott so much shit over him aborting his simulation prematurely.

In the exact same spirit - why isn’t the answer of his decider “approximate” ?

> >
> >> And ∞ + 1 isn't a defined value. The problem being that if you try to
> >> treat ∞ just like any other number, you get into all sorts of contradictins.
> > That is just a straw man. I am not treating it as a number any more than you are treating it as a number when you put it on top of Σ.
> >
> > We can just treat the range() function as polymorphic. It can range over [0, end] for finite values, or [0, ∞) for infinite values.
> >
> >>>> So the problem is YOU are treating it as part of a continuum that it
> >>>> isn't part of.
> >>> What? Yes YOU are treating it as part of the continuum! Not me!
> >>>
> >>> [0, ∞) UNION ∞ is [0,∞]
> >> except that [0, ∞) and ∞ are different sorts of things..
> > Yes, and? Your UNION operator doesn’t seem to mind the type mismatch?
> Union was an Operator on TYPES, creating the TYPE (which might not have
> a name) that represents something that can be a Real or an Infinte.
Yes, R is a sub-type of infinite types!

The type “infinity” already contains the type R!

> > So what is the result of the operation UNiON([0, ∞), ∞) ?
> UNiON([0, ∞), ∞), if you let it take that sort of thing.
>
> I defined UNION on TYPES, not ranges, but it could be.
If you can’t reason in type theory let’s reason in set theory.

Does infinity contain the set of Reals?

> >>>
> >> But MY system doesn't have a contiuum of [0, ∞]
> > Then what do you mean by [0, ∞) UNION ∞ ?
> The combination of the Set of Real Nubers from 0 to an unbounded
> maximum, unioned the infinite value ∞. it doesn't need to be able to be
> reduced.
Ffs. It is like talking to a brick wall.

If you are treating infinity as a value then it is no different to the value of number 5!

> >
> >> or a parametric overload of the limit operatior, so it can't be my system.
> > Ohhh! So you are using a different system to Ben? You guys should have a chat!
> >
> > He says lim() is overloaded for asymptotic and non-asymptotic limits.
> Right, it just isn't a parameteric overload. non-asymptotic limits, i.e.
> limits to a real value, have one definition, and asymptotic limits, i.e.
> limits to an infinite value. have a different (but related) definition.
But different. They key word here is “differentiation”.

How do you differentiate one kind of limit from the other kind of limit?

> >> All that is an effet of you trying to redefine things into a different
> >> fundamental framework.
> > I am not redefining anything. I am UNIFYING your mess into a framework that deals with polymorphism.
> >
> > Generic types.
> SO do so. If you can't, that is YOUR problem for not using a powerful
> enough system.
Idiot. My system is more powerful than yours.

I can treat infinity as a value without special pleading.

> That would be like signing up to the Indy 500 with a Go-Kart.
No, it is like trying to explain to an Indy 500 racer (you) that I am in a time machine.

> >> Any problem there-of, are YOUR problems, not mine.
> > Your mess is everyone’s problem.
> >
> It is well defined. If you can't translate that to your system, then
> your system isn't powerful enough.
Ahhh, the irony!

“Well defined” is well defined in my system.
It is not well defined in your system.

Well-defined is self-defined. There is never any ambiguity when a system means exactly itself.

> >
> >> I know how to take a limit, and how to define it.
> > You don’t know shit. You can’t even define “define”, but you think you can define a limit.
> >
> >> The fact that you have
> >> issues converting from a system based on limits to one where you are
> >> bound by computability is your own problem, not mine.
> > It is difficult to explain to an idiot that he doesn’t understand.
> >
> > I am no more bound by computability than you are bound by your “foundation”.
> >
> > I am merely demonstrating that my foundation is more foundational.
> It seems your foundation doesn't HAVE a foundation to stand on.
Neither does yours.

But my unfounded foundation is more powerful than your unfounded foundation..

On 8/31/22 5:56 AM, Skep Dick wrote:
> On Tuesday, 30 August 2022 at 14:52:37 UTC+2, richar...@gmail.com wrote:
>> On 8/30/22 6:52 AM, Skep Dick wrote:
>>> On Tuesday, 30 August 2022 at 02:28:32 UTC+2, richar...@gmail.com wrote:
>>>> On 8/29/22 8:21 AM, Skep Dick wrote:
>>>>> On Monday, 29 August 2022 at 14:12:35 UTC+2, richar...@gmail.com wrote:
>>>>>> So, since they are not defined as parameterized, maybe you system can't
>>>>>> handle it.
>>>>> Shame! Are you tripping up over syntax?
>>>>>
>>>>> sigma(start, end, f)
>>>>>
>>>>> In [1]: def sigma(start, end, f):
>>>>> ...: sum = 0
>>>>> ...: for x in range(start, end + 1):
>>>>> ...: sum = sum + f(x)
>>>>> ...: return sum
>>>>>
>>>>> In [2]: sigma(0,2, (lambda x:x))
>>>>> Out[2]: 3
>>>>>
>>>>> In [3]: sigma(0,2, (lambda x: 2**x))
>>>>> Out[3]: 7
>>>> yes, but that program won't work with end = ∞, as it will never finish.
>>> I think you are lying. The program will work just fine, but it will never finish.
>>> It is just an infinite loop.
>>>
>>> The value of “sum” inside the program will continue to accumulate.
>> If you want ot call that "work", you never get an answer, not even an
>> approximat one.
>
> Of course I call it work. You don’t? An approximate answer is incomplete work!
> At ANY point of the execution you can sample the value of “sum”. So you can always know the value after n

But you have no idea how "close" the answer is, or if it will eventually
settle down to a number.

THAT is a key difference. With Limit Theory, you can only approximate if
you can bound your error, and show that you can get an answer for ANY
arbitrary small error.

>
> Surely you of all people would understand? You are giving Olcott so much shit over him aborting his simulation prematurely.
>
> In the exact same spirit - why isn’t the answer of his decider “approximate” ?

Because there is no answer between Halting and Non-Halting, so there is
no degree of being right. THe answer is just wrong.

>
>
>>>
>>>> And ∞ + 1 isn't a defined value. The problem being that if you try to
>>>> treat ∞ just like any other number, you get into all sorts of contradictins.
>>> That is just a straw man. I am not treating it as a number any more than you are treating it as a number when you put it on top of Σ.
>>>
>>> We can just treat the range() function as polymorphic. It can range over [0, end] for finite values, or [0, ∞) for infinite values.
>>>
>>>>>> So the problem is YOU are treating it as part of a continuum that it
>>>>>> isn't part of.
>>>>> What? Yes YOU are treating it as part of the continuum! Not me!
>>>>>
>>>>> [0, ∞) UNION ∞ is [0,∞]
>>>> except that [0, ∞) and ∞ are different sorts of things.
>>> Yes, and? Your UNION operator doesn’t seem to mind the type mismatch?
>> Union was an Operator on TYPES, creating the TYPE (which might not have
>> a name) that represents something that can be a Real or an Infinte.
> Yes, R is a sub-type of infinite types!
>
> The type “infinity” already contains the type R!

No it doesn't, at least not the one I am talking about. It has two
values +∞ (sometimes just noted as ∞) and -∞, but not any of the value
in between.

>
>>> So what is the result of the operation UNiON([0, ∞), ∞) ?
>> UNiON([0, ∞), ∞), if you let it take that sort of thing.
>>
>> I defined UNION on TYPES, not ranges, but it could be.
> If you can’t reason in type theory let’s reason in set theory.
>
> Does infinity contain the set of Reals?

No.

>
>
>>>>>
>>>> But MY system doesn't have a contiuum of [0, ∞]
>>> Then what do you mean by [0, ∞) UNION ∞ ?
>> The combination of the Set of Real Nubers from 0 to an unbounded
>> maximum, unioned the infinite value ∞. it doesn't need to be able to be
>> reduced.
> Ffs. It is like talking to a brick wall.
>
> If you are treating infinity as a value then it is no different to the value of number 5!
>

It is different to the value 5 in that The Real Number system defines
only VERY limited operations on it. Mainly that you can take a limit to
it, using a special definition of the limit function.

>
>
>
>
>>>
>>>> or a parametric overload of the limit operatior, so it can't be my system.
>>> Ohhh! So you are using a different system to Ben? You guys should have a chat!
>>>
>>> He says lim() is overloaded for asymptotic and non-asymptotic limits.
>> Right, it just isn't a parameteric overload. non-asymptotic limits, i.e.
>> limits to a real value, have one definition, and asymptotic limits, i.e.
>> limits to an infinite value. have a different (but related) definition.
> But different. They key word here is “differentiation”.
>
> How do you differentiate one kind of limit from the other kind of limit?

By the type of the value we are going "to".

Is it a Finite value, or an Infinite Value.

>
>>>> All that is an effet of you trying to redefine things into a different
>>>> fundamental framework.
>>> I am not redefining anything. I am UNIFYING your mess into a framework that deals with polymorphism.
>>>
>>> Generic types.
>> SO do so. If you can't, that is YOUR problem for not using a powerful
>> enough system.
> Idiot. My system is more powerful than yours.
>
> I can treat infinity as a value without special pleading.

Only if you are willing to break your system. Once you let ∞ - ∞ = 0 all
sorts of problem crop up.

There are lots of examples of this, your not understanding them shows
you lack of foundation.

>
>> That would be like signing up to the Indy 500 with a Go-Kart.
> No, it is like trying to explain to an Indy 500 racer (you) that I am in a time machine.
>
>>>> Any problem there-of, are YOUR problems, not mine.
>>> Your mess is everyone’s problem.
>>>
>> It is well defined. If you can't translate that to your system, then
>> your system isn't powerful enough.
> Ahhh, the irony!
>
> “Well defined” is well defined in my system.
> It is not well defined in your system.
>
> Well-defined is self-defined. There is never any ambiguity when a system means exactly itself.
>
>>>
>>>> I know how to take a limit, and how to define it.
>>> You don’t know shit. You can’t even define “define”, but you think you can define a limit.
>>>
>>>> The fact that you have
>>>> issues converting from a system based on limits to one where you are
>>>> bound by computability is your own problem, not mine.
>>> It is difficult to explain to an idiot that he doesn’t understand.
>>>
>>> I am no more bound by computability than you are bound by your “foundation”.
>>>
>>> I am merely demonstrating that my foundation is more foundational.
>> It seems your foundation doesn't HAVE a foundation to stand on.
> Neither does yours.
>
> But my unfounded foundation is more powerful than your unfounded foundation.
>
>

Nope, it has collased, and you don't realize it.

On 8/31/22 2:58 AM, wij wrote:
> On Tuesday, 30 August 2022 at 19:38:38 UTC+8, skepd...@gmail.com wrote:
>> On Tuesday, 30 August 2022 at 12:36:13 UTC+2, Andy Walker wrote:
>>> On 30/08/2022 01:05, Ben Bacarisse wrote:
>>>> Skep Dick <skepd...@gmail.com> writes:
>>>>> So the fact that every student of yours abandons the epsilon-delta
>>>>> method the moment they cross ocer into a Physics lecture was no signal
>>>>> of importance to you?
>>>> If they did (and I have no idea if they did) why would I care?
>>> Probably unlike Ben, I spent a significant part of my career
>>> teaching maths to physicists. I never tried to teach them delta-
>>> epsilon analysis. In the UK, basic calculus is done at school, so
>>> they came to university with a decent basic knowledge already of
>>> how to differentiate and integrate. It was more important to teach
>>> them the more advanced things they needed /for physics/. Even more
>>> important to teach them the limits of what they knew -- "If /this/
>>> happens, you need to consult a mathematician." Much the same with
>>> engineers, and others who needed post-school mathematics.
>>>>> The fact that Physicists (to this day) use calculus with
>>>>> infinitesimals is of no importance to you?
>>> Not in my modules they didn't.
>>>> I'm interested in infinitesimals. I like infinitesimals. I wish the
>>>> 0.999... = 1 deniers knew how to use them.
>>> Likewise. But they're not a good way to do calculus. You
>>> either run into all the problems that Newton, Euler, ... ran into in
>>> the 18thC and which led to rigorous analysis in the 19thC, or you
>>> have to work with hyperreals or surreals or similar, and that's an
>>> awful lot of extra baggage to carry around for something so simple
>>> [for most physics/engineering purposes]. It's more fruitful to
>>> teach limits informally, and emphasise that this is for "well-
>>> behaved functions" *only*. Good and bad behaviour illustrated by
>>> examples rather than by formal definitions.
>>>
>>> --
>>> Andy Walker, Nottingham.
>>> Andy's music pages: www.cuboid.me.uk/andy/Music
>>> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Handel
>> Take the usual way for calculating the derivative of f(x)=x^2
>>
>> f’(x) =(f(x +dx) - f(x)) / dx = …… = 2x + dx
>>
>> At this point we say stuff like “dx is so small we can basically ignore it” and conclude f’(x)=2x
>>
>> But who really buys this argument? If dx is so small that we can ignore it then ignore it sooner…
>>
>> f’(x) =(f(x +dx) - f(x)) / dx = (f(x) - f(x))/dx = 0
>
> I think I solved this problem.
> The basic problem of calculus is the definition of slope of a tangent line.
> slope= (f(x+h)-f(x))/h .... (h is ?)
> The issue is that tangent lines require h be zero (otherwise, it is a secant line
> defined by two points), which causes diving zero problems.
>
> My idea is:
> Let g(x,h)= (f(x+h)-f(x))/h
>
> g'(x,h)≡ identical to g(x,h) and g'(x,0) is defined (h is defined in the set of infinitesimal).
> Slope of tangent line is defined as g'(x,0). There can only be at most one version of g'.
> [note] g' and g are looked very close (only different in one point), but the
> essence are very different. g' is defined on a deleted set.
>
> Result: Power functions have such g' (exact) slope function, therefore, usual
> x^2,x^3,.. geometric analyses are exactly correct.
> Exponential and trigonometric functions are questionable (but the error is infinitesimal).
> And probably many concepts like 'continuous'... developed in the limit theory
> and 'real analysis' can be saved. All begin from a definition of '∞'.
>
> ---
> The use of "dy/dx" is different, probably some kind of logic. I said nothing about this.

You don't seem to understand how limit theory works.

Yes, for any finite value slope = (f(x+h) - F(x))/h only gives an
approximate answer.

But when we use the limit

slope = lim(h->0) (f(x+h) - f(x))/h

we get the exact tangent provided that f(x) meets the requirement that
it is continuous a x, which is one of the requriements to compute the
derivative.