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  "body": "\n## Statement\n\nLet $a \u003c c \u003c b$ and let $f \\colon (a, b) \\to \\mathbb{R}$ be increasing.\nThen both [[one-sided limits|0909PWZR62]] of $f$ at $c$ exist, and\n$$\\lim_{x \\to c^-} f(x) = \\sup_{a \u003c x \u003c c} f(x) \\le f(c) \\le \\inf_{c \u003c x \u003c b} f(x) = \\lim_{x \\to c^+} f(x).$$\n\n## Remarks\n\nThe proof is the same as for the [[monotone convergence theorem|0FM6N4R0VB]]. The supremum exists because $f(c)$ is an upper bound.\nA similar statement holds for decreasing functions, with the inequalities reversed.\nSo $f$ is continuous at $c$ exactly when the two one-sided limits are equal. Otherwise $c$ is a [[jump discontinuity|05B0FKFXQA]].\nMathlib states the left limit as `Monotone.tendsto_nhdsLT`.\n\n## Explanations\n\n### L1 human\n\nAn increasing function can only move up as $x$ approaches $c$ from the left.\nIts values are bounded by $f(c)$, so they settle at a limit.\n",
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