Bounded derivative Riemann integrable

In summary, the conversation discusses the relationship between differentiability, boundedness, and Riemann integrability of a function. The two given facts suggest that a function being differentiable and bounded does not necessarily imply that its derivative is Riemann integrable. This is supported by the example of Volterra's function. The speaker also mentions attempting to come up with a counterexample involving x^2sin(1/x), but was unsuccessful.
  • #1
jostpuur
2,116
19
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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  • #2
jostpuur said:
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
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!

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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FAQ: Bounded derivative Riemann integrable

What is a bounded derivative?

A bounded derivative refers to a function whose derivative is always within a certain range of values. In other words, the slope of the function never becomes too steep or too flat.

What does it mean for a function to be Riemann integrable?

A function is Riemann integrable if it can be approximated by a series of rectangles whose areas converge to a specific value as the size of the rectangles decreases.

How is bounded derivative related to Riemann integrability?

A bounded derivative is a necessary condition for a function to be Riemann integrable. Without a bounded derivative, the function's slope may change too drastically for the area under the curve to be accurately approximated.

Can a function have a bounded derivative but not be Riemann integrable?

Yes, it is possible for a function to have a bounded derivative but not be Riemann integrable. This can happen if the function has discontinuities or infinite oscillations that prevent it from being approximated by rectangles.

How is the bounded derivative Riemann integrable property useful in real-world applications?

The bounded derivative Riemann integrable property is useful in many scientific and engineering applications, such as calculating the work done by a force, finding the center of mass of an object, and estimating the volume of an irregularly shaped object. It allows us to accurately approximate complex functions and solve real-world problems.

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