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  "body": "\n## Summary\n\nA functional equation is an equation whose unknown is a [[function|0EPAT51DXQ]] and which must hold for all values of some variables.\nBoth sides are built by composing the unknown function with known operations.\nClassic examples are [[Cauchy's functional equation|ZBMEVWN100]] $f(x + y) = f(x) + f(y)$ and the rule $\\Gamma(x + 1) = x\\,\\Gamma(x)$ of the gamma function.\nBy convention, equations with derivatives or integrals of the unknown are called differential or integral equations instead.\nFunctional equations characterize many classical functions, such as the exponential, the logarithm and the power functions, by a few simple rules.\n\n## Explanations\n\n### L1 human\n\nIn school algebra the unknown is a number; in a functional equation the unknown is a whole function.\nOne looks for every function that obeys a rule for all inputs.\nFor example, the rule $f(x + y) = f(x) f(y)$ says that adding inputs multiplies outputs.\nThe continuous solutions on the real line are $f = 0$ and $f(x) = a^x$ with $a \u003e 0$, so the rule pins down the exponential functions.\n",
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