Relation between function being Riemann integrable and continuous and piecewise smooth

Question

Is it equivalent to say that $\vert f^2\vert$ is Riemann integrable and it is continuous and piecewise smooth? I don't seem to be able to find any counter example.

Answer 1

This is not true, take for instance

$$f(x)=\begin{cases} 0 &\text{ if $x\in [0,1)$}\\ 1 &\text{ if $x\in [1,2]$}.\end{cases}$$

Then $f$ is not continuous, but $\vert f^2\vert$ is Riemann integrable.