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  "body": "\n## Statement\n\nLet $f \\colon \\R \\to \\R$ be an [[additive function|KF719SZD5T]], that is, $f(x + y) = f(x) + f(y)$ for all $x, y \\in \\R$.\nAssume that there are real numbers $a \u003c b$ and $M$ such that $f(x) \\le M$ for all $x$ in the [[interval|0Q1D85B8G6]] $(a, b)$.\nThen $f(x) = f(1)\\, x$ for all $x \\in \\R$.\n\n## Remarks\n\nSet $c = f(1)$ and $g(x) = f(x) - cx$.\nThen $g$ is additive and $g(q) = 0$ for every rational $q$, so $g(x + q) = g(x)$ for all real $x$ and rational $q$.\nEvery real $x$ has a rational $q$ with $x + q \\in (a, b)$, and $g$ is bounded above on $(a, b)$, so $g$ is bounded above on all of $\\R$.\nIf $g(x_0) \\ne 0$ for some $x_0$, then $g(n x_0) = n g(x_0)$ or $g(-n x_0) = -n g(x_0)$ grows without bound as $n \\to \\infty$, a contradiction.\nSo $g = 0$ and $f(x) = cx$.\nThe same holds if $f$ is bounded below on an interval, since then $-f$ is bounded above.\nWithout any bound the conclusion fails: [[additive functions built from a Hamel basis|FFTNS66XG6]] are not linear.\n\n## Explanations\n\n### L1 human\n\nAn additive function is either a straight line through the origin or a wild function.\nThis theorem says that a wild additive function cannot stay below a fixed level even on a tiny interval.\nSo a single bound on a small piece of the line already forces the function to be linear.\n\n### L2 human\n\nThis is the basic regularity criterion for [[Cauchy's functional equation|ZBMEVWN100]].\nContinuity at a point, monotonicity on an interval and measurability all imply boundedness on some interval, so they all reduce to this result.\nFor measurable functions this reduction goes through the Steinhaus theorem; see [[measurable solutions of Cauchy's equation|TWANGP59RB]].\nThe same idea shows that the graph of a non-linear additive function is [[dense in the plane|MNAWWP9J1X]].\n",
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  "name": "Darboux's theorem on additive functions bounded on an interval",
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