Injective continuous real-valued functions on an interval must be monotone
It is a well-known and very useful theorem of real analysis that injective continuous real-valued functions on an interval must be monotone. This is usually proved by contradiction using the Intermediate Value Theorem (IVT). However the direct negation of a function being monotone is actually not very friendly. Let $I$ be a (nondegenerate) interval in $\R$. A function $f:I\to\R$ is monotone increasing if (and only if), for all $x_1,x_2 \in I$ with $x_1\leq x_2$, we have $f(x_1) \leq f(x_2)$. Negating this we see that $f$ is not monotone increasing if there exist $x_1,x_2 \in I$ with $x_1\leq x_2$ such that $f(x_1) > f(x_2)$. [Of course in this case we must actually have $x_1<x_2$.] Similarly, $f$ is not monotone decreasing if and only if there exist $x_3,x_4 \in I$ with $x_3\leq x_4$ such that $f(x_3) < f(x_4)$. [Again, in this case we must have $x_3<x_4$.] Now $f$ is monotone (on $I$) if and only if $f$ is monotone increasing or $f$...