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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]], let $f \\colon X \\to [-\\infty, \\infty]$ be a function to the [[extended real line|0GSZ7B0CGR]], and let $x_0 \\in X$.\n\n1. The function $f$ is *upper semicontinuous at* $x_0$ if for each real $y \u003e f(x_0)$ there is a [[neighbourhood|01T3TMGTTE]] $U$ of $x_0$ with $f(x) \u003c y$ for all $x \\in U$.\n2. The function $f$ is *lower semicontinuous at* $x_0$ if for each real $y \u003c f(x_0)$ there is a neighbourhood $U$ of $x_0$ with $f(x) \u003e y$ for all $x \\in U$.\n3. The function $f$ is *upper* (or *lower*) *semicontinuous* if it is upper (or lower) semicontinuous at each point of $X$.\n\n## Remarks\n\nFor $X \\subseteq \\R$, upper semicontinuity at $x_0$ means $\\limsup_{x \\to x_0} f(x) \\le f(x_0)$, and lower semicontinuity means $\\liminf_{x \\to x_0} f(x) \\ge f(x_0)$.\nThe function $f$ is upper semicontinuous if and only if the set $\\set{x \\in X : f(x) \u003c y}$ is open for each real $y$.\nIt is lower semicontinuous if and only if $\\set{x \\in X : f(x) \u003e y}$ is open for each real $y$.\nThe function $f$ is upper semicontinuous if and only if $-f$ is lower semicontinuous.\nA real function is [[continuous|0P27W9RW0G]] at $x_0$ if and only if it is both upper and lower semicontinuous at $x_0$.\n\nExample: the [[indicator function|0VSB0XF4G7]] $f = \\mathbf 1_{[0, 1]}$ on $\\R$ is upper semicontinuous.\nThe set $\\set{f \u003c y}$ is empty for $y \\le 0$, is $\\R \\setminus [0, 1]$ for $0 \u003c y \\le 1$, and is $\\R$ for $y \u003e 1$, and each of these is open.\n\nNon-example: the same function is not lower semicontinuous at $0$.\nTake $y = 1/2 \u003c 1 = f(0)$; each neighbourhood of $0$ contains a point $x \u003c 0$ with $f(x) = 0 \u003c y$.\n",
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