Bounded derivative Riemann integrable

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  • #1
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Assume that a function [itex]f:[a,b]\to\mathbb{R}[/itex] is differentiable in all points of its domain, and that the derivative [itex]f':[a,b]\to\mathbb{R}[/itex] 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.
 
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Answers and Replies

  • #2
pasmith
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Assume that a function [itex]f:[a,b]\to\mathbb{R}[/itex] is differentiable in all points of its domain, and that the derivative [itex]f':[a,b]\to\mathbb{R}[/itex] is bounded. Is the derivative necessarily Riemann integrable?

No: Volterra's function is a counterexample.
 
  • #3
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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 [itex]x^2\sin (\frac{1}{x})[/itex] as basis! :surprised

But I was unable to get a counter example working.

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