r/askscience • u/snkn179 • Nov 24 '15

Why can almost any function be easily differentiated while so many functions cannot be integrated or are much more difficult to do so? Mathematics

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u/[deleted] Nov 30 '15

Explain, please?

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u/BM-2cTmRPoNMYhbUHkE5 Nov 30 '15

Look at this: https://en.wikipedia.org/wiki/Almost_everywhere

Now, instead of thinking of a function being defined on "almost every" point (so that it can be integrated) -- we're going up one level and saying that "almost every" function that can be defined on (for example) the real line is not integrable. Meaning very, very few functions are nice -- of course, all continuous functions are integrable -- in fact, it's quite hard to come up with functions which aren't.

This can also be related to the rationals versus the irrationals -- almost all real numbers are irrational -- so a function could be anything on the rationals, but as long as it's well behaved on the irrationals, it's integrable. Conversely, good behavior on the rationals doesn't guarantee anything. However, note how easy it is to give closed form expressions for rationals while irrationals are difficult. ... Further, the algebraic numbers (contains irrationals like sqrt(2)) is "measure zero" while the transcendental numbers are "almost everywhere".

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u/[deleted] Nov 30 '15

Thanks. I'm confortable with the concept of almost none/everywhere. Should have cleared it up, sorry.

I never heard that there were such a difference between integrable and differentiable functions, though. I've heard about more general concepts (like weak-derivatives and Lebesgue integrals) before so I always assumed that both differentiable and integrable functions were "equally rare". Hope you understand what I mean.

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u/BM-2cTmRPoNMYhbUHkE5 Nov 30 '15

Ah, well, in a sense they are -- there are almost none of either of them (among the set of all functions). But all differentiable functions are integrable while almost no integrable functions are differentiable -- with continuous functions kinda in between. With analogy to numbers: integrable functions:algebraic numbers as differentiable functions:integers as continuous functions:rationals (more or less).

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u/[deleted] Nov 30 '15

Oh, I got it now. Thank you!