A Continuous Injective Function on a Closed Interval is Strictly Monotone

In the setting of Single-Variable Calculus on an Interval, let $a,b\in\mathbb{R}$ with $a<b$, let $[a,b]$ be the closed interval determined by $a$ and $b$, and let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and injective, that is,

$$f(x)\ne f(y)\qquad\text{whenever }x,y\in[a,b]\text{ and }x\ne y .$$

Problem. Show that $f$ is strictly monotone on $[a,b]$.