Bounded derivative Riemann integrable

[jostpuur]

Assume that a function $f:[a,b]\to\mathbb{R}$ is differentiable in all points of its domain, and that the derivative $f':[a,b]\to\mathbb{R}$ is bounded. Is the derivative necessarily Riemann integrable?

This what I know:

Fact 1: Assume that a function is differentiable at all points of its domain. Then the derivative is not necessarily Riemann integrable.

Fact 2: Assume that a function is bounded. Then the function is not necessarily Riemann integrable.

So my question is not obvious.

 

[pasmith]

Assume that a function $f:[a,b]\to\mathbb{R}$ is differentiable in all points of its domain, and that the derivative $f':[a,b]\to\mathbb{R}$ is bounded. Is the derivative necessarily Riemann integrable?

No: Volterra's function is a counterexample.

 

[jostpuur]

It is unfortunate that I cannot prove to you my honesty, but I swear that I came up with this question on my own, and was also attempting to construct a counter example with the $x^2\sin (\frac{1}{x})$ as basis!

But I was unable to get a counter example working.

Actually I think I'm also unable to understand the explanation on Wikipedia page.