Bounded derivative Riemann integrable

  • Thread starter jostpuur
  • Start date
  • #1
2,111
17

Main Question or Discussion Point

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.
 
Last edited:

Answers and Replies

  • #2
pasmith
Homework Helper
1,738
410
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
2,111
17
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.
 
Last edited:

Related Threads on Bounded derivative Riemann integrable

  • Last Post
Replies
19
Views
6K
  • Last Post
Replies
13
Views
3K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
8
Views
3K
  • Last Post
2
Replies
28
Views
2K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
5
Views
2K
Replies
30
Views
8K
Top