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

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 ${f}^{\prime }\left(x\right)>0$ on $\mathbb{R}$ or ${f}^{\prime }\left(x\right)<0$ on $\mathbb{R}$).
The claim is definitely true if ${f}^{\prime }s$ domain were instead a closed interval, but since $\mathbb{R}$ is open, I don't understand why Marsden's claim should be true.
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Kali Galloway
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...