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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]], let $f \\colon X \\to [-\\infty, +\\infty]$ be a function with values in the [[extended real line|0GSZ7B0CGR]], and let $x_0 \\in X$.\n\n1. $f$ is *upper semicontinuous* at $x_0$ if for every real $y \u003e f(x_0)$ there is a [[neighbourhood|0ZCY0Z80B6]] $U$ of $x_0$ with $f(x) \u003c y$ for all $x \\in U$.\n2. $f$ is *lower semicontinuous* at $x_0$ if for every real $y \u003c f(x_0)$ there is a neighbourhood $U$ of $x_0$ with $f(x) \u003e y$ for all $x \\in U$.\n\nThe function $f$ is upper (lower) semicontinuous if it is upper (lower) semicontinuous at every point of $X$.\n\n## Remarks\n\nThe function $f$ is upper semicontinuous if and only if $\\set{x : f(x) \u003c y}$ is [[open|02BH3MCBQB]] for every real $y$.\nIt is lower semicontinuous if and only if $\\set{x : f(x) \u003e y}$ is open for every real $y$.\nIn terms of limits, $f$ is upper semicontinuous at $x_0$ if and only if $\\limsup_{x \\to x_0} f(x) \\le f(x_0)$, and lower semicontinuous at $x_0$ if and only if $\\liminf_{x \\to x_0} f(x) \\ge f(x_0)$.\nA real-valued $f$ is [[continuous|0VGFHTM75Y]] at $x_0$ if and only if it is both upper and lower semicontinuous at $x_0$.\n\nExample: the [[floor function|08GR54CKEA]] $\\floor{\\cdot} \\colon \\R \\to \\R$ is upper semicontinuous.\nAt an integer $n$ and for $y \u003e n$, the neighbourhood $(n - 1, n + 1)$ works, since $\\floor{x} \\le n \u003c y$ there.\nAt a non-integer, the floor function is locally constant.\n\nNon-example: the indicator function $f$ of $[0, 1]$ in $\\R$ is not lower semicontinuous at $0$.\nHere $f(0) = 1$; for $y = 1/2$, every neighbourhood of $0$ contains some $x \u003c 0$ with $f(x) = 0 \u003c y$.\n",
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