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  "body": "\n## Statement\n\nLet $(S, +)$ be a [[semigroup|0XMEM406JM]], let $\\delta \\ge 0$, and let $f \\colon S \\to \\C$ satisfy\n$$\\abs{f(x + y) - f(x) f(y)} \\le \\delta \\quad \\text{for all } x, y \\in S.$$\nThen one of the following holds:\n\n1. $\\abs{f(x)} \\le \\frac{1 + \\sqrt{1 + 4\\delta}}{2}$ for all $x \\in S$;\n2. $f(x + y) = f(x) f(y)$ for all $x, y \\in S$.\n\n## Remarks\n\nWrite $K = \\frac{1 + \\sqrt{1 + 4\\delta}}{2}$, so that $K^2 - \\delta = K$ and $K \\ge 1$.\nIf $\\abs{f(a)} = K + \\eta$ with $\\eta \u003e 0$, then $\\abs{f(2a)} \\ge (K + \\eta)^2 - \\delta \\ge K + 2\\eta$, and iterating shows that $f$ is unbounded.\nIf $f$ is unbounded, compare $f(x + y + z)$ with $f(x + y) f(z)$ and with $f(x) f(y + z)$ to get\n$$\\abs{f(x + y) - f(x) f(y)} \\abs{f(z)} \\le 2\\delta + \\abs{f(x)} \\delta,$$\nand let $\\abs{f(z)} \\to \\infty$.\nSo an approximate solution of [[Cauchy's exponential equation|CEGF8W64VQ]] is either bounded or an exact solution.\nThis is called superstability: it is stronger than [[Hyers–Ulam stability|TPYXQTYND6]], since an unbounded approximate solution must be exact.\nExample: on $S = (\\R, +)$, the constant $f = 1/2$ satisfies the hypothesis with $\\delta = 1/4$, and it is bounded by $K = (1 + \\sqrt{2})/2$ but not exponential.\nThe result was first proved by J. Baker, J. Lawrence and F. Zorzitto in 1979 for real functions on a vector space.\n\n## Explanations\n\n### L1 human\n\nSuppose a function satisfies $f(x + y) = f(x) f(y)$ up to a fixed error.\nEither the function stays small everywhere, or the error is in fact zero.\nA large value of $f$ would magnify any error without bound, which the fixed bound forbids.\n\n### L2 human\n\nFor the additive Cauchy equation, approximate solutions are near exact ones but need not be exact.\nThe exponential equation behaves differently because it is multiplicative: an error is amplified by large values of $f$.\nSimilar superstability results hold for d'Alembert's equation and other multiplicative equations.\n",
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