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  "body": "\n## Definition\n\n*Cauchy's exponential equation* is the [[functional equation|SRAN4EKWS0]]\n$$f(x + y) = f(x)\\, f(y) \\quad \\text{for all } x, y \\in \\R,$$\nfor an unknown [[function|0EPAT51DXQ]] $f \\colon \\R \\to \\R$.\nA solution is a function $f$ that satisfies this identity for all real $x$ and $y$.\n\n## Remarks\n\nExample: $f(x) = e^{3x}$ is a solution, since $e^{3(x + y)} = e^{3x} e^{3y}$.\nNon-example: $f(x) = 1 + x$ is not a solution, since $f(1 + 1) = 3$ while $f(1) f(1) = 4$.\nEvery solution satisfies $f(x) = f(x/2)^2 \\ge 0$.\nIf $f(x_0) = 0$ for one $x_0$, then $f(x) = f(x - x_0) f(x_0) = 0$ for all $x$.\nOtherwise $f \u003e 0$ and $\\ln f$ is an [[additive function|KF719SZD5T]], so the solutions are $f = 0$ and $f = e^{A}$ with $A$ additive.\nThe continuous solutions are $f = 0$ and $f(x) = a^x$ with $a \u003e 0$.\n",
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