# 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 function that is monotonic on every closed interval is monotonic on the whole of $\mathbb R$. – TonyK Nov 29 '13 at 9:35
• Otherwise, $f'$ would take both a positive value and a negative value. There is a closed interval containing the points at which they occur. So... – David Mitra Nov 29 '13 at 9:42
• @DavidMitra Ah! Thanks, would you post this as an answer? – Ryan Nov 29 '13 at 9:49

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...