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  "body": "\n## Definition\n\nLet $D \\subseteq \\mathbb{R}^n$, let $f \\colon D \\to \\mathbb{R}^m$, and let $a \\in D$.\nThe function $f$ is continuous at $a$ when, for each $\\varepsilon \u003e 0$, there is a $\\delta \u003e 0$ such that $\\|f(x) - f(a)\\| \u003c \\varepsilon$ for each $x \\in D$ with $\\|x - a\\| \u003c \\delta$.\nHere $\\|\\cdot\\|$ is the Euclidean norm of [[Euclidean space|0XPR1E85W0]].\nThe function $f$ is continuous on $D$ when it is continuous at each point of $D$.\n\n## Remarks\n\nLet $a$ be a limit point of $D$. Then $f$ is continuous at $a$ if and only if $\\lim_{x \\to a} f(x) = f(a)$, with the [[limit of a function of several variables|0J8Y1NCY0K]].\nA function into $\\mathbb{R}^m$ is continuous if and only if each component function is continuous.\nSums, products and compositions of continuous functions are continuous. So each polynomial in $x_1, \\dots, x_n$ is continuous on $\\mathbb{R}^n$.\nContinuity in each variable separately is weaker: see the [[separately continuous function that is not continuous|0C0KPAKK9Y]].\n\n## Explanations\n\n### L1 human\n\nA continuous function has no jumps: if you move the input a little, the output moves only a little.\nIn the plane, \"a little\" means inside a small disc, so the condition must hold for moves in every direction.\n",
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