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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[metric spaces|063W5JMY85]] and let $f \\colon X \\to Y$ be a function.\nThen $f$ is [[continuous|0SY9ECQ19Q]] if and only if $f(x_n) \\to f(x)$ for each sequence $(x_n)$ in $X$ that [[converges|06JYPJ4WHH]] to a point $x$.\n\n## Remarks\n\nThe condition on sequences is sequential continuity.\nProof of the converse: if $f$ is not continuous at $x$, then by [[epsilon-delta continuity|0EKGKY5TGH]] there are $\\varepsilon \u003e 0$ and points $x_n$ with $d(x_n, x) \u003c 1/n$ and $d(f(x_n), f(x)) \\ge \\varepsilon$. Then $x_n \\to x$, but $f(x_n) \\not\\to f(x)$.\nThe same equivalence holds at each single point.\nIn general topological spaces, continuity implies sequential continuity, but the converse can fail.\n\n## Explanations\n\n### L1 human\n\nA continuous map sends convergent sequences to convergent sequences, with the right limit.\nTo show that a map is not continuous, it is often enough to find one sequence $x_n \\to x$ with $f(x_n) \\not\\to f(x)$.\n",
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  "msc": [
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