I'm working on the following as part of a larger proof and am struggling with this last piece of the puzzle.
Let $f$ be differentiable on an interval $A$. If $f'(x)$ is never $0$ on $A$, then prove either $f'(x)>0$ or $f'(x)<0$ for all $x$ in $A$.
I think if I can solve this bit then I can solve the next piece which is to say that $f$ is monotone on $A$ and then ultimately that $f$ is one to one on $A$.
Thank you!