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  "body": "\n## Statement\n\nLet $D \\subseteq \\mathbb{R}$, let $f \\colon D \\to \\mathbb{R}$, and let $a \\in D$.\nThen $f$ is [[continuous at|0EEE2CZB2A]] $a$ if and only if the following holds.\nFor each $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ with $|f(x) - f(a)| \u003c \\varepsilon$ for all $x \\in D$ with $|x - a| \u003c \\delta$.\n\n## Remarks\n\nEach open interval around a point contains an interval of the form $(c - r, c + r)$ with $r \u003e 0$. This gives the equivalence.\nMany texts take this condition as the definition of continuity.\nMathlib states a form for metric spaces as `Metric.continuousAt_iff`.\n\n## Explanations\n\n### L1 human\n\nFor each allowed error $\\varepsilon$ in the output, there is a tolerance $\\delta$ in the input.\nInputs within $\\delta$ of $a$ give outputs within $\\varepsilon$ of $f(a)$.\n",
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