Q:

Using the intermediate value theorem for derivatives to infer that a function is strictly monotonic

My textbook Elementary Classical Analysis claims that by Darboux's theorem (the intermediate value theorem for derivatives), if a function $f:\mathbb R\to\mathbb R$ has a nonzero derivative on $\mathbb R$, then is $f$ strictly monotonic (i.e., either $f'(x)>0$ on $\mathbb R$ or $f'(x)<0$ on $\mathbb R$).
The claim is definitely true if $f's$ domain were instead a closed interval, but since $\mathbb R$ is open, I don't understand why Marsden's claim should be true.

A:

Argue by contradiction.  If $f$ is not strictly monotonic, then, $f′$ would take both a positive value and a negative value. There is a closed interval containing the points at which these values occur. So... 

