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  "body": "\n## Statement\n\nLet $I \\subseteq \\R$ be an open [[interval|0Q1D85B8G6]], and let $f \\colon I \\to \\R$ be [[midpoint convex|1TQ1JQXEWP]].\nAssume that $f$ is bounded above on some nonempty open interval $J \\subseteq I$.\nThen $f$ is [[continuous|0X3WGX95FE]] on $I$, and $f$ is [[convex|07Q7CNM5WF]].\n\n## Remarks\n\nThe idea of the proof has three steps.\nFirst, midpoint convexity and induction give $f(qx + (1 - q)y) \\le q f(x) + (1 - q) f(y)$ for all dyadic rationals $q \\in [0, 1]$, and this carries the upper bound from $J$ to a neighbourhood of every point of $I$.\nSecond, an upper bound $M$ near a point $x_0$ gives a lower bound there, from $f(x_0) \\le \\frac{f(x_0 + h) + f(x_0 - h)}{2}$.\nThird, with $f$ locally bounded, the inequality with $q = 2^{-n}$ gives $\\abs{f(x_0 + 2^{-n} h) - f(x_0)} \\le 2^{-n} C$ for small $h$, which yields continuity; a continuous midpoint convex function is convex.\n\nThe boundedness hypothesis cannot be dropped.\nA discontinuous additive function $a \\colon \\R \\to \\R$, built from a Hamel basis, is midpoint convex but not continuous, and it is unbounded above on every interval.\nExample of use: a midpoint convex function on $(0, 1)$ with values in $[0, 1]$ is automatically convex.\nThe theorem holds in the same form for midpoint convex functions on an open convex subset of $\\R^n$.\n\n## Explanations\n\n### L1 human\n\nMidpoint convexity is a weak form of convexity that only looks at midpoints.\nOn its own it allows very wild functions.\nThe theorem says that the wildness disappears as soon as the function is bounded above on one small interval: then the function is continuous and truly convex.\n\n### L2 human\n\nThe theorem is the convexity analogue of the classical results on Cauchy's equation, where an additive function bounded on an interval is linear.\nIt reduces many regularity questions to finding an upper bound on some interval.\nFor instance, Sierpiński's theorem obtains such a bound for measurable functions, and so shows that measurable midpoint convex functions are convex.\n",
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