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  "body": "\n## Statement\n\nLet $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]], let $f \\colon X \\to Y$ be a function and let $a \\in X$.\nThen $f$ is [[continuous at|07N7W5C29X]] $a$ if and only if for each real $\\varepsilon \u003e 0$ there is a real $\\delta \u003e 0$ such that\n$$d_X(x, a) \u003c \\delta \\implies d_Y\\bigl(f(x), f(a)\\bigr) \u003c \\varepsilon \\quad \\text{for all } x \\in X.$$\n\n## Remarks\n\nIn terms of [[open balls|0DC3MPR7C9]], the condition says: for each $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ with $f(B(a, \\delta)) \\subseteq B(f(a), \\varepsilon)$.\nThe number $\\delta$ can depend on $\\varepsilon$ and on $a$. When one $\\delta$ works for all $a$, the map is [[uniformly continuous|0FHSDCQACY]].\nFor $X = Y = \\mathbb{R}$, this is the epsilon-delta definition of continuity from calculus.\n\n## Explanations\n\n### L1 human\n\nYou can make $f(x)$ as close to $f(a)$ as you want by taking $x$ close enough to $a$.\nThe tolerance $\\varepsilon$ for the output comes first, and then you find a tolerance $\\delta$ for the input.\n",
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