Does the definite integral of a function's derivative exists?

Asked May 04 '17 at 08:03

Active May 04 '17 at 11:53

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Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be differentiable such that $f(0)=0$ and $f(1)=1$ and let $|f'(x)|\le 2$ for all real $x$.

Is the derivative integrable? I know that the derivative needn't be integrable, but if it's bounded can it be so? What is the condition for a derivative to be integrable? (I mean Riemann-Integrable)

edited May 04 '17 at 11:53

asked May 04 '17 at 08:03

It's evidently Lebesgue integrable, since it's measureable and bounded. Indicate what you mean by integrability. – Michał Miśkiewicz May 04 '17 at 09:02
thanks for pointing out; Riemann Integrable. – May 04 '17 at 11:54
Whether it's a derivative or not, a bounded function $g \colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if the set of its discontinuities is a (Lebesgue) null set. – Daniel Fischer May 04 '17 at 12:43
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See here for some information about how badly behaved a derivative can be. – Daniel Fischer May 04 '17 at 21:21

Does the definite integral of a function's derivative exists?

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