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I know Newton-Leibnetz theorem:
if $f \in \mathcal{R}[a ,b]$($f$ is Riemann integrable on $[a ,b]$), and if exists differentiable function $F$ satisfy $F'=f$ on $[a, b]$, then $$\int_{a}^{b}f(x)dx=F(b)-F(a).$$ And I wonder if $F$ is Riemann integrable, whether or not $F'=f$ is Riemann integrable? I can't see it directly from Newton-Leibnetz theorem since it firstly acquires $f$ to be Riemann integrable, which is what I want to prove.

I referred this, but it was talking about $|f|$ is integrable.

Edit: if $F'$ is not, how to make it is Riemann integrable? Such as making $F$ be differentiable over $[a, b]$?

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Ryan
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  • $F$ being Riemann integrable doesn't imply $F$ is differentiable at any point. Indeed there are continuous, nowhere differentiable functions, and every continuous function is Riemann integrable. – Ian Dec 15 '14 at 03:13
  • I have editted my question, but I guess that letting $F$ be differentiable over $[a, b]$ is still not enough. Am I right? @Ian – Ryan Dec 15 '14 at 03:28
  • You are correct. There are functions which are differentiable with bounded derivative yet the derivative is not Riemann integrable. E.g. Volterra's function. – kahen Dec 15 '14 at 04:54

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