prove the convexity of zero one loss multiple a convex function I am now working on a problem To prove the convexity of a zero-one loss multiple with a convex function, and it looks like this: $$L(s) = s^2 \times \boldsymbol 1(s\leq 0);$$ when proving this convexity, is it ok to assume this $\boldsymbol 1(s\leq 0)$ a constant?
if it is not ok to treat 1(s<=0) as constant, how would we prove the convexity then?

Note that
$$
L'(s)=
\begin{cases}
2s&,s \leq0
\\
0 &,s>0
\end{cases}
$$
What do you know about a differentiable function whose derivative is monotonically increasing?
