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  "body": "\n## Statement\n\nLet $I$ be an interval and let $f \\colon I \\to \\mathbb{R}$ be continuous and injective.\nThen $f$ is [[strictly monotone|0TR9SR9FTB]].\n\n## Remarks\n\nSuppose $x \u003c y \u003c z$ in $I$ with $f(y)$ not between $f(x)$ and $f(z)$. Then the [[intermediate value theorem|00KAFBR32E]] gives a value of $f$ taken twice, which contradicts injectivity.\nContinuity is needed. The function on $[0, 2]$ with $f(x) = x$ for $x \u003c 1$ and $f(x) = 3 - x$ for $x \\ge 1$ is injective but not monotone.\nMathlib states a form for closed intervals as `ContinuousOn.strictMonoOn_of_injOn_Icc`.\n\n## Explanations\n\n### L1 human\n\nA continuous function on an interval that goes up and then down would take some value twice.\nSo if it never repeats a value, it must always go up or always go down.\n",
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