Fundamental Theorem of Calculus

mchinen 34 minutes ago | parent | next [-]

I've studied the proofs before but there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative at only two points, especially for wobbly non monotonic functions.

I feel similar about the trace of a matrix being equal to the sum of eigenvalues.

Probably this means I should sit with it more until it is obvious, but I also kind of like this feeling.

	▲	ironSkillet 22 minutes ago \| parent \| next [-]
		It is not determined by the derivative, it's the antiderivative, as someone else mentioned. The derivative is the rate of change of a function. The "area under a curve" of the graph of a function measures how much the function is "accumulating", which is intuitively a sum of rates of change (taken to an infinitesimal limit).
	▲	sambapa 29 minutes ago \| parent \| prev [-]
		You meant antiderivative?

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emacdona 36 minutes ago | parent | prev | next [-]

> f is Riemann integrable iff it is bounded and continuous almost everywhere.

FWIW, I think this is the same as saying "iff it is bounded and has finite discontinuities". I like that characterization b/c it seems more precise than "almost everywhere", but I've heard both.

I mention that because when I read the first footnote, I thought this was a mistake:

> boundedness alone ensures the subinterval infima and suprema are finite.

But it wasn't. It does, in fact, insure that infima and suprema are finite. It just does NOT ensure that it is Riemann integrable (which, of course the last paragraph in the first section mentions).

Thanks for posting. This was a fun diversion down memory lane whilst having my morning coffee.

If anyone wants a rabbit hole to go down:

Think about why the Dirichlet function [1], which is bounded -- and therefore has upper and lower sums -- is not Riemann integrable (hint: its upper and lower sums don't converge. why?)

Then, if you want to keep going down the rabbit hole, learn how you _can_ integrate it (ie: how you _can_ assign a number to the area it bounds) [2]

[1] One of my favorite functions. It seems its purpose in life is to serve as a counter example. https://en.wikipedia.org/wiki/Dirichlet_function

[2] https://en.wikipedia.org/wiki/Lebesgue_integral

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mjdv 30 minutes ago | parent | next [-]

> 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 have Lebesgue measure 0, including all countable sets.

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emacdona 24 minutes ago | parent [-]

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?

	▲	jfarmer 14 minutes ago \| parent \| next [-]
		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...
	▲	ironSkillet 19 minutes ago \| parent \| prev \| next [-]
		No that's not true either. A quick Google will reveal many examples, in particular the "Cantor set".
	▲	thaumasiotes 20 minutes ago \| parent \| prev [-]
		The Cantor set is uncountable and has Lebesgue measure 0.

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jfarmer 30 minutes ago | parent | prev | next [-]

"Almost everywhere" means "everywhere except on a set of measure 0", in the Lebesgue measure sense.

Here's an example of a Riemann integrable function w/ infinitely many discontinuities: https://en.wikipedia.org/wiki/Thomae%27s_function

Anyone interested in this should check out the Prologue to Lebesgue's 1901 paper: http://scratchpost.dreamhosters.com/math/Lebesgue_Integral.p...

It gives several reasons why we "knew" the Riemann integral wasn't capturing the full notion of integral / antiderivative

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bandrami 27 minutes ago | parent | prev | next [-]

"Almost everywhere" is precisely defined, and it is broader than that. E.g. the real numbers are almost everywhere normal, but there are uncountably many non-normal numbers between any two normal reals.

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sambapa 25 minutes ago | parent | prev | next [-]

"almost everywhere" can mean the curve has countably infinite number of discontinuities

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Jaxan 29 minutes ago | parent | prev [-]

“Almost everywhere” is a mathematical term and can mean two things (I think):

- except finitely many, or

- except a set of measure zero.

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mellosouls an hour ago | parent | prev | next [-]

https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculu...

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bikrampanda an hour ago | parent | prev [-]

What is the font used on the site?

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viscousviolin an hour ago | parent | next [-]

Today I learned there's a CSS property for styling the first letter of a paragraph, neat. (https://css-tricks.com/almanac/properties/i/initial-letter/)

--edit: The font used for those initials is called Goudy Initialen: https://www.dafont.com/goudy-initialen.font

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crispyambulance an hour ago | parent | prev | next [-]

That font, and how it's integrated with the math looks amazing. Katex for the math?

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viscousviolin an hour ago | parent [-]

Seems like Katex from the scripts getting loaded. I love the design too, kinda medieval-chic.

	▲	eru 40 minutes ago \| parent [-]
		Looks more early modern to me. :)

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emmelaich an hour ago | parent | prev | next [-]

Inspection suggests "https://online-fonts.com/fonts/alegreya"

	▲	thaumasiotes 41 minutes ago \| parent [-]
		Also https://online-fonts.com/fonts/goudy-initialen

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genezeta an hour ago | parent | prev [-]

Alegreya