If $f$ is differentiable in $[a,b]$ and $f'(x) \ne 0$ for each $x \in [a,b]$ so $f$ is monotone in $[a,b]$? If $f$ is differentiable in $[a,b]$ and $f'(x) \ne 0$ for each $x \in [a,b]$ so $f$ is monotone in $[a,b]$.
Is this correct? I don't think so. Because the differntial doesn't have to be continuous so it can be positive in one area and negative in other... but I can't find a counter-example...
thanks

By Darboux's theorem, we have that $f'$ is a Darboux function, i.e. maps full intervals to full intervals, in other words, it guarantees that the range of $f'$,
$f'([a,b])$ is a full interval which now avoids $0$, hence it lies on either the positive or the negative ray.
