Exactly one of the requests is impossible. Provide examples for the other three. Assume functions $f$ and $g$ are defined on all of R.
(i) Functions $f$ and $g$ not differentiable at zero but where $fg$ is differentiable at $0$.
(ii) A function $f$ is not differentiable at zero and a function $g$ differentiable at zero where $fg$ is differentiable at $0$.
My answer: let $f=sin(1/x)$ and $g=0$
(iii) A function $f$ not differentiable at zero and a function $g$ differentiable at zero where $f+g$ is differentiable at zero.
(iv) A function $f$ differentiable at zero but not differentiable anywhere else.
I'm not looking for the answers, just looking for some guidance. I have been trying to think of examples for a while but still cannot get anything. Closest I have came is trying to create a piecewise continous function using Weierstrass for Part (iv) so that it can be differentiable at $0$. Still haven't figured it out though.
Best Answer
Let's get wild here.
For (i): Define $f$ to be $0$ on $(-\infty,0],$ define $g$ to be $0$ on $[0,\infty)$ and define these functions to be a complete disaster everywhere else.
(ii) What is the easiest, flattest, most stupid differentiable function $g$ on $\mathbb {R}$ that you know?
(iii) $f = (f+g) - g.$
(iv) Note that if $|f(x)|\le x^2$ for all $x,$ then $f'(0)=0.$ That leaves a lot of room for $f$ to misbehave elsewhere.