Two non-differentiable functions whose product is differentiable. So I was wondering while studying analysis if there is any case where two functions aren't differential at $0$ (kind of like $1/x$) but is differentiable at 0 when combined (i.e.  $fg$).
I mean this for functions that are defined on $\mathbb{R}$.

On $\mathbb R$ let $f$ be the indicator of $[0,\infty)$ and $g$ the indicator of $(-\infty,0)$. Their product is $0$, whereas each of the functions is not differentiable at $0$.
Furthermore, you dont need your functions to be defined on $\mathbb R$ to say something about differentiability in $0$, since differentiability is a local property.
