Injective functions have no functional extrema Is it true that all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that are injective (one-to-one) have no functional extrema? I can't find any counterexample, but that can easily be proved when we have a monotonic function.

Here is a specific example. Take
$$
f(x) = \left\{\begin{array}{ll}
   \arctan(x) & \mbox{if $x \neq \pm 1$} \\
   \pi/2 & \mbox{if $x = -1$} \\
   -\pi/2 & \mbox{if $x = 1$}
\end{array}\right.
$$
