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 (https://en.wikipedia.org/wiki/Darboux%27s_theorem_%28analysis%29), every derivative satisfies the intermediate value theorem (even though it may not be continuous).
Hence, either $f' > 0$ or $f' < 0$ for all $x$.
