If $h:[a,b]\to R$ is continuous and one-to-one, then $h$ is monotone.

Proof: The image of the connected set {(s,t): a <= s < t <= b} under the map $h(t)-h(s)$ is a connected subset of $R\setminus\{0\}$.

If $h:[a,b]\to R$ is continuous and one-to-one, then $h$ is monotone.

Proof: The image of the connected set $\{(s,t): a \le s < t \le b\}$ under the map $h(t)-h(s)$ is a connected subset of $R\setminus\{0\}$.