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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]].\nLet $\\Gamma$ be a [[multivalued function|03CPD1HGF3]] from $X$ to $Y$, that is, a map that sends each $x \\in X$ to a nonempty subset $\\Gamma(x) \\subseteq Y$.\nLet $x_0 \\in X$.\n\n1. $\\Gamma$ is *upper hemicontinuous* at $x_0$ if the following holds for every [[open|02BH3MCBQB]] set $V \\subseteq Y$ with $\\Gamma(x_0) \\subseteq V$.\n   There is a [[neighbourhood|0ZCY0Z80B6]] $U$ of $x_0$ with $\\Gamma(x) \\subseteq V$ for all $x \\in U$.\n2. $\\Gamma$ is *lower hemicontinuous* at $x_0$ if the following holds for every open set $V \\subseteq Y$ with $\\Gamma(x_0) \\cap V \\ne \\emptyset$.\n   There is a neighbourhood $U$ of $x_0$ with $\\Gamma(x) \\cap V \\ne \\emptyset$ for all $x \\in U$.\n\n$\\Gamma$ is upper (lower) hemicontinuous if it is so at every point of $X$, and *continuous* if it is both.\n\n## Remarks\n\nSome authors say upper and lower semicontinuous instead of hemicontinuous.\nIf $\\Gamma(x) = \\set{f(x)}$ for a function $f \\colon X \\to Y$, then each of the two conditions is equivalent to the [[continuity|0VGFHTM75Y]] of $f$ at $x_0$.\n\nExample: let $\\Gamma \\colon \\R \\to \\R$ with $\\Gamma(0) = [0, 1]$ and $\\Gamma(x) = \\set{0}$ for $x \\ne 0$.\nIt is upper hemicontinuous at $0$: an open $V \\supseteq [0, 1]$ contains $0$, so $\\Gamma(x) \\subseteq V$ for all $x$.\n\nNon-example: the same $\\Gamma$ is not lower hemicontinuous at $0$.\nThe open set $V = (1/2, 2)$ meets $\\Gamma(0)$, but $\\Gamma(x) \\cap V = \\emptyset$ for every $x \\ne 0$.\n",
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