TITLE: A continuous, injective function $f: \mathbb{R} \to \mathbb{R}$ is either strictly increasing or strictly decreasing.
QUESTION [27 upvotes]: I would like to prove the statement in the title.
Proof: We prove that if $f$ is not strictly decreasing, then it must be strictly increasing.
So suppose $x < y$.
And that's pretty much how far I got.  Help will be appreciated.

REPLY [19 votes]: Consider $g\colon \{(x,y)\mid x<y\}\mapsto\mathbb R$, defined by $g(x,y):=f(x)-f(y)$. Clearly $g$ is continuos.  Since the domain of $g$ is connected and $g$ has no zeroes, the image of $g$ is an interval not containing $0$.