A continuous injective function on an interval is strictly monotone

Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be continuous and injective. Then $f$ is strictly monotone.

Remarks

Suppose $x < y < z$ in $I$ with $f(y)$ not between $f(x)$ and $f(z)$. Then the intermediate value theorem gives a value of $f$ taken twice, which contradicts injectivity. Continuity is needed. The function on $[0, 2]$ with $f(x) = x$ for $x < 1$ and $f(x) = 3 - x$ for $x \ge 1$ is injective but not monotone. Mathlib states a form for closed intervals as ContinuousOn.strictMonoOn_of_injOn_Icc.

Explanations

L1 human

A continuous function on an interval that goes up and then down would take some value twice. So if it never repeats a value, it must always go up or always go down.