Example of a function f that is Generalized Riemann Integrable, but its square is NOT Generalized Riemann Integrable.

Question

I am reading a section about Generalized Riemann Integral (Kurzweil-Henstock), and there was a problem on that section to provide an example of a function $f$ on $[0,1]$ that is Generalized Riemann Integrable, but its square $f^2$ is NOT Generalized Riemann Integrable.

I couldn't come up with a single function that does the job. Therefore, I appreciate if anyone can provide me with an example of a function $f$ that satisfies the conditions mentioned above. Thanks!

Answer 1

Consider $f(x)=\frac{1}{\sqrt{x}}$ if $x\ne 0$ and any value if $x=0.$