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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]], let $f \\colon X \\to Y$ be a function and let $a \\in X$.\nThe function $f$ is continuous at $a$ when for each [[neighbourhood|09CJ00H3NW]] $V$ of $f(a)$, the [[preimage|0FGCFY1AZX]] $f^{-1}(V)$ is a neighbourhood of $a$.\n\n## Remarks\n\nEquivalently, for each neighbourhood $V$ of $f(a)$ there is a neighbourhood $U$ of $a$ with $f(U) \\subseteq V$.\nA function is a [[continuous map|0SY9ECQ19Q]] if and only if it is continuous at each point of $X$.\nFor metric spaces, continuity at $a$ is the same as the [[epsilon-delta condition at a|0EKGKY5TGH]].\n\n## Explanations\n\n### L1 human\n\nContinuity at $a$ means that you can keep $f(x)$ as close to $f(a)$ as you want, if you keep $x$ close enough to $a$.\nNeighbourhoods replace the distances of the metric case.\n",
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