I am having trouble with this:
If $h(x)=0$ for $x<0$ and $h(x)=1$ for $x\geq 0$, prove there does not exist a function $f:\mathbb{R}\to \mathbb{R}$ such that $f'(x)=h(x)$ for all $x \in \mathbb{R}$. Give examples of two functions, not differing by a constant, whose derivatives equal $h(x)$ for all $x\neq 0$.
For the last part, Obviously if $h(x) = 1$ for $x<0$ and $h(x)=x$ for $x>0$ it will work. Is there another example? Also, I am not sure where to begin on the proof.