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.
Continuous Injective Function – Strictly Increasing or Decreasing
calculusreal-analysis
Best Answer
Prove the contrapositive instead: if $f$ is not strictly increasing and not strictly decreasing, then it is not one-to-one.
For example, say there are points $a\lt b\lt c$ such that $f(a)\lt f(b)$ and $f(b)\gt f(c)$. Either $f(a)=f(c)$ (in which case $f$ is not one-to-one), or $f(a)\lt f(c)$, or $f(c)\lt f(a)$.
If $f(a)\lt f(c)\lt f(b)$, then by the Intermediate Value Theorem there exists $d\in (a,b)$ such that $f(d)=f(c)$; hence $f$ is not one-to-one.
Now, there are other possibilities (I made certain assumptions along the way, and you should check what the alternatives are if they are not met).