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