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  "body": "\n## Statement\n\nLet $X$ be a [[topological space|0CNQXVBHCX]], let $K \\subseteq X$ be a nonempty [[compact set|00R0X1WGPX]] and let $f \\colon X \\to \\mathbb{R}$ be a [[continuous map|0SY9ECQ19Q]].\nThen there are points $a, b \\in K$ with $f(a) \\le f(x) \\le f(b)$ for all $x \\in K$.\n\n## Remarks\n\nProof: the set $f(K)$ is compact in $\\mathbb{R}$, because the [[continuous image of a compact set is compact|0K9AAJ8G51]]. By the [[Heine–Borel theorem|07PFYV65MB]], it is closed and bounded. Thus it contains its [[supremum|0BTMQ04556]] and its infimum.\nFor $K = [a, b]$, this is the extreme value theorem of calculus.\nCompactness is needed: $f(x) = x$ on $(0, 1)$ has no maximum, and $f(x) = x$ on $\\mathbb{R}$ is not bounded.\n\n## Explanations\n\n### L1 human\n\nA continuous function on a compact set has a largest value and a smallest value, and it reaches them at some points.\nOn an open interval, the function can come closer and closer to a value without reaching it.\n",
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