The space $SC_{2\pi}:=SC_{2\pi}(\Bbb R,\Bbb K)$ is defined as the set of piecewise continuous functions with period $2\pi$, that is, if $f\in SC_{2\pi}$ then $f|_{[0,2\pi]}$ have a finite number of jump-discontinuities.
And if $f,g\in SC_{2\pi}$ we can define it convolution as
$$(f*g)(x):=\frac1{2\pi}\int_0^{2\pi}f(x-y)g(y)\mathrm dy$$
The space $C_{2\pi}$ is the set of continuous $2\pi$-periodic functions. I have an exercise that says
Show that if $f,g\in C_{2\pi}$ then $f*g\in C_{2\pi}$.
What I did is just the observation that by the periodicity of $f$ then
$$(f*g)(x):=\frac1{2\pi}\int_0^{2\pi}f(x-y)g(y)\mathrm dy=\frac1{2\pi}\int_0^{2\pi}f(x+2\pi-y)g(y)\mathrm dy$$
hence $f*g\in C_{2\pi}$. But for this observation I dont used the continuity of $f$ or $g$ so I think that we can drop it and show the same for any pair $f,g\in SC_{2\pi}$.
Then my question is: we can drop the condition that $f$ and $g$ are continuous to just let them piecewise continuous functions? I dont see a reason to think the contrary.