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Fundamental Theorem of Calculus

david.alvarezrosa.com

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Re: Fundamental Theorem of Calculus

#21

Post by thaumasiotes »

emacdona wrote: post #19
mjdv wrote: post #13

Earlier quoted context omitted.

> FWIW, I think this is the same as saying "iff it is bounded and has finite discontinuities". It is not: for example, the piece-wise constant function f: [0,1] -> [0,1] which starts at f(0) = 0, stays constant until suddenly f(1/2) = 1, until f(3/4) = 0, until f(7/8) = 1, etc. is Riemann integrable. "Continuous almost everywhere" means that the set of its discontinuities has Lebesgue measure 0. Many infinite sets ha…

Ah, thanks for the clarification! Would it have been accurate then to have said: "iff it is bounded and has countable discontinuities"? Or, are there some uncountable sets which also have Lebesgue measure 0?

The Cantor set is uncountable and has Lebesgue measure 0.

Re: Fundamental Theorem of Calculus

#22

Post by ironSkillet »

emacdona wrote: post #19
mjdv wrote: post #13

Earlier quoted context omitted.

> FWIW, I think this is the same as saying "iff it is bounded and has finite discontinuities". It is not: for example, the piece-wise constant function f: [0,1] -> [0,1] which starts at f(0) = 0, stays constant until suddenly f(1/2) = 1, until f(3/4) = 0, until f(7/8) = 1, etc. is Riemann integrable. "Continuous almost everywhere" means that the set of its discontinuities has Lebesgue measure 0. Many infinite sets ha…

Ah, thanks for the clarification! Would it have been accurate then to have said: "iff it is bounded and has countable discontinuities"? Or, are there some uncountable sets which also have Lebesgue measure 0?

No that's not true either. A quick Google will reveal many examples, in particular the "Cantor set".

Re: Fundamental Theorem of Calculus

#23

Post by jfarmer »

emacdona wrote: post #19
mjdv wrote: post #13

Earlier quoted context omitted.

> FWIW, I think this is the same as saying "iff it is bounded and has finite discontinuities". It is not: for example, the piece-wise constant function f: [0,1] -> [0,1] which starts at f(0) = 0, stays constant until suddenly f(1/2) = 1, until f(3/4) = 0, until f(7/8) = 1, etc. is Riemann integrable. "Continuous almost everywhere" means that the set of its discontinuities has Lebesgue measure 0. Many infinite sets ha…

Ah, thanks for the clarification! Would it have been accurate then to have said: "iff it is bounded and has countable discontinuities"? Or, are there some uncountable sets which also have Lebesgue measure 0?

No, it's really sets of measure zero. The Cantor set is an example of an uncountable set of measure 0: https://en.wikipedia.org/wiki/Cantor_set

The indicator function of the Cantor set is Riemann integrable. Like you said, though, the Dirichlet function (which is the indicator function of the rationals) is not Riemann integrable.

The reason is because the Dirchlet function is discontinuous everywhere on [0,1], so the set of discontinuities has measure 1. The Cantor function is discontinuous only on the Cantor set.

Likewise, the indicator function of a "fat Cantor set" (a way of constructing a Cantor-like set w/ positive measure) is not Riemann integrable: https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%9...

Re: Fundamental Theorem of Calculus

#25

Post by shmoil »

Good job, David. Have a lollipop. Now learn & write up the proof that the Henstock-Kurzweil integral integrates _every_ derivative. This is what we had in my calculus class on top of the outdated Riemann integral.

Re: Fundamental Theorem of Calculus

#26

Post by EdwardDiego »

> This post introduces the Riemann integral

Sweet! I'm keen to learn about the basic fundamentals of calculus!

> For each subinterval ...(bunch of cool maths rendering I can't copy and paste because it's all comes out newline delimited on my clipboard) ... and let mk and Mk denote the infimum and supremum of f on that subinterval...

Okay, guess it wasn't the kind of introduction I had assumed/hoped.

Very cool maths rendering though.

As someone who never passed high school or got a degree thanks to untreated ADHD, if anyone knows of an introduction to the basic fundamentals of calculus that a motivated but under educated maths gronk can grok, I would gratefully appreciate a link or ten.

Re: Fundamental Theorem of Calculus

#27

Post by moi2388 »

EdwardDiego wrote: post #26

> This post introduces the Riemann integral Sweet! I'm keen to learn about the basic fundamentals of calculus! > For each subinterval ...(bunch of cool maths rendering I can't copy and paste because it's all comes out newline delimited on my clipboard) ... and let m k and M k denote the infimum and supremum of f on that subinterval... Okay, guess it wasn't the kind of introduction I had assumed/hoped. Very cool maths…

Khan academy

Re: Fundamental Theorem of Calculus

#28

Post by homeonthemtn »

EdwardDiego wrote: post #26

> This post introduces the Riemann integral Sweet! I'm keen to learn about the basic fundamentals of calculus! > For each subinterval ...(bunch of cool maths rendering I can't copy and paste because it's all comes out newline delimited on my clipboard) ... and let m k and M k denote the infimum and supremum of f on that subinterval... Okay, guess it wasn't the kind of introduction I had assumed/hoped. Very cool maths…

https://en.wikipedia.org/wiki/Calculus_Made_Easy#:~:text=Cal...

1910 book, but actually does the job well

Re: Fundamental Theorem of Calculus

#29

Post by dnemmers »

EdwardDiego wrote: post #26

> This post introduces the Riemann integral Sweet! I'm keen to learn about the basic fundamentals of calculus! > For each subinterval ...(bunch of cool maths rendering I can't copy and paste because it's all comes out newline delimited on my clipboard) ... and let m k and M k denote the infimum and supremum of f on that subinterval... Okay, guess it wasn't the kind of introduction I had assumed/hoped. Very cool maths…

https://calculusmadeeasy.org/1.html

Re: Fundamental Theorem of Calculus

#30

Post by mr_mitm »

EdwardDiego wrote: post #26

> This post introduces the Riemann integral Sweet! I'm keen to learn about the basic fundamentals of calculus! > For each subinterval ...(bunch of cool maths rendering I can't copy and paste because it's all comes out newline delimited on my clipboard) ... and let m k and M k denote the infimum and supremum of f on that subinterval... Okay, guess it wasn't the kind of introduction I had assumed/hoped. Very cool maths…

Yeah, judging by the terseness, this is clearly aimed at undergrads. Then again, this is covered in literally every calculus class, so I'm not sure who this is supposed to be for.
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