5
$\begingroup$

The function $f:[-1,1] \to \mathbb{R}$ given by $f(x) = x^2\sin\left(\frac{1}{x^2}\right)$ is an example of a function whose derivative is not Lebesgue integrable on $[-1,1]$ but is improper Riemann integrable.

Volterra constructed a function $f:[0,1] \to \mathbb{R}$ whose derivative is bounded (hence Lebesgue integrable) but not Riemann integrable (proper or improper).

Is there a differentiable function $f:[a,b] \to \mathbb{R}$ such that $f'(x)$ is neither improper Riemann integrable nor Lebesgue integrable on $[a,b]$?

Such a derivative is Henstock–Kurzweil integrable , since all derivatives happen to be Henstock–Kurzweil integrable.

The example here unfortunately does not work. It has jump discontinuities, hence isn't a derivative.

$\endgroup$
10
$\begingroup$

Combine your first example on one interval with your second on another.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.