Does every continuous injection of real numbers preserve or reverse order?

Question

Let $f:(a,b)\to\mathbb R$ be a continuous injection. Must $f$ be order-preserving ($x<y\Rightarrow f(x)<f(y)$) or order-reversing ($x<y\Rightarrow f(x)>f(y)$)?

Answer 1

Let $x<y<z$.

• If $f(x)<f(z)<f(y)$ or $f(y)<f(z)<f(x)$, then by the intermediate value theorem, there is some $c\in(x,y)$ with $f(c)=f(z)$, violating the fact that $f$ is injective.

• If $f(y)<f(x)<f(z)$ or $f(z)<f(x)<f(y)$, then by the intermediate value theorem, there is some $c\in(y,z)$ with $f(c)=f(x)$, violating the fact that $f$ is injective.

Thus $f(x)<f(y)<f(z)$ or $f(z)<f(y)<f(x)$.