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  "body": "\n## Definition\n\nLet $X$ be a set and let $f \\colon X \\to [-\\infty, +\\infty]$ be a function with values in the [[extended real line|0GSZ7B0CGR]].\nThe *epigraph* of $f$ is the subset of the [[Cartesian product|09CEDJYBKP]] $X \\times \\R$ given by\n$$\\operatorname{epi} f = \\set{(x, r) \\in X \\times \\R : f(x) \\le r}.$$\n\n## Remarks\n\nOnly real numbers $r$ are used, so points $x$ with $f(x) = +\\infty$ contribute nothing, and points with $f(x) = -\\infty$ contribute the whole line $\\set{x} \\times \\R$.\nThe *strict epigraph* uses $f(x) \u003c r$ instead.\nIf $X$ is a real vector space, $f$ is convex if and only if $\\operatorname{epi} f$ is a convex set.\nIf $X$ is a topological space, $f$ is lower semicontinuous if and only if $\\operatorname{epi} f$ is closed in $X \\times \\R$.\nThe function is recovered as $f(x) = \\inf \\set{r \\in \\R : (x, r) \\in \\operatorname{epi} f}$, with $\\inf \\emptyset = +\\infty$.\n\nExample: for $f(x) = x^2$ on $\\R$, the point $(1, 2)$ lies in $\\operatorname{epi} f$, since $f(1) = 1 \\le 2$.\n\nNon-example: the point $(2, 1)$ does not lie in $\\operatorname{epi} f$, since $f(2) = 4 \u003e 1$.\n",
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