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 $\mathbb{R}$.
 A: Take $f(x)=|x|$ and $g(x)=|x|$.
A: Let $h :R\to R$ be positive and differentiable at every point. Example 1.Let $f:R\to R$ be positive and nowhere continuous. For example let $f(x)=1$ when $x$ is rational and $f(x)=2$ when $x$ is rational.  Example 2. Let $f:R\to R$ be continuous ,positive, and nowhere differentiable.(Such functions do exist.)...In both examples let $g(x)=h(x)/f(x)$. In both examples , neither $f$ nor $g$ are differentiable anywhere but their product is differentiable everywhere. In example 2., both $f$ and $g$ are continuous. For a third example suppose $f(x)=0$ for $x\leq 0$ and that $f$ is discontinuous on $[0,\infty)$. Let $g(x)=f(-x)$. 
A: 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.
