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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $k \\ge 0$ be an integer or $k = \\infty$.\nLet $f \\colon \\R^n \\to \\C$ be locally integrable, that is, integrable on every compact set.\nLet $g \\colon \\R^n \\to \\C$ be a $C^k$ function with compact support.\nThen the [[convolution|2Z29JN3VES]] $f * g$ is a $C^k$ function on $\\R^n$, and $\\partial^\\alpha (f * g) = f * \\partial^\\alpha g$ for every multi-index $\\alpha$ with $\\abs{\\alpha} \\le k$.\nIn particular, if $g$ is a [[smooth function|0V33BNKZ9S]] with compact support, then $f * g$ is smooth.\n\n## Remarks\n\nFor $k = 0$, continuity of $f * g$ follows from the uniform continuity of $g$ and the integrability of $f$ on compact sets.\nThe case $k \\ge 1$ follows by induction from the [[derivative of a convolution|9FN0PFZZXZ]], applied to the $C^{k-1}$ functions $\\partial_j g$.\nThe case $k = \\infty$ is the conjunction of all finite cases.\n\nExample: $f = \\mathbf{1}_{[0, 1]}$ is discontinuous on $\\R$.\nIf $0 \u003c \\varepsilon \u003c 1/2$ and $\\varphi$ is a smooth function with support in $[-\\varepsilon, \\varepsilon]$ and integral $1$, then $f * \\varphi$ is smooth, equals $1$ on $[\\varepsilon, 1 - \\varepsilon]$ and $0$ outside $[-\\varepsilon, 1 + \\varepsilon]$.\n\nThe smoothness comes only from $g$: if $g$ is merely continuous, $f * g$ need not be differentiable.\nFor $f = g = \\mathbf{1}_{[0, 1]}$, neither factor is continuous, yet $f * g$ is the continuous tent function, which is not differentiable at $0$, $1$, $2$.\n\n## Explanations\n\n### L1 human\n\nAveraging a rough function against a smooth bump produces a smooth function.\nThe result is at least as smooth as the bump, however rough the original function was.\nThis is how blurring removes sharp edges from a picture.\n\n### L2 human\n\nWith a [[mollifier|CQMK4YPE1J]] $\\varphi_\\varepsilon$, the functions $f * \\varphi_\\varepsilon$ are smooth and tend to $f$ in $L^p$ as $\\varepsilon \\to 0$ when $f \\in L^p$, $1 \\le p \u003c \\infty$.\nThis is the standard way to approximate integrable or continuous functions by smooth ones.\nIt is used to prove density theorems, to define weak derivatives through smooth approximations, and to regularize distributions.\n",
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