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  "body": "\n## Summary\n\nA [[functional equation|SRAN4EKWS0]] is Hyers–Ulam stable if each approximate solution is near an exact one.\nPrecisely, a function that satisfies it up to an error $\\delta$ lies within $K\\delta$ of a solution, for a fixed constant $K$.\nThe notion comes from a question of Ulam on approximate homomorphisms.\n[[Hyers' theorem|VFRJ0P2P15]] shows that the Cauchy equation $f(x+y) = f(x) + f(y)$ is stable, with $K = 1$, for maps from a normed space to a Banach space.\nStability results justify replacing an approximate solution by an exact one.\n\n## Explanations\n\n### L1 human\n\nMeasurements and computations are never exact, so a function found in practice satisfies an equation only roughly.\nStability says this is harmless: a rough solution is near a true solution.\nFor example, $f(x) = x + \\sin x$ satisfies $f(x + y) = f(x) + f(y)$ up to an error at most $3$, and it stays within distance $1$ of the true solution $x \\mapsto x$.\nSome equations are not stable, and then a rough solution can drift arbitrarily far from every true one.\n",
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