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  "body": "\n## Definition\n\nLet $p$ be a nonzero real number.\nThe *periodicity equation* with period $p$ is the [[functional equation|SRAN4EKWS0]]\n$$f(x + p) = f(x) \\quad \\text{for all } x \\in \\R,$$\nfor an unknown [[function|0EPAT51DXQ]] $f \\colon \\R \\to Y$, where $Y$ is a set.\nThe solutions are the [[periodic functions|0ATF6Q0633]] with period $p$.\n\n## Remarks\n\nExample: $f(x) = \\cos(\\pi x)$ is a solution for $p = 2$, since $\\cos(\\pi x + 2\\pi) = \\cos(\\pi x)$.\nNon-example: $f(x) = x^2$ is not a solution for any $p \\ne 0$, since $f(0 + p) = p^2 \\ne 0 = f(0)$.\nWith $q = \\abs{p}$, the general solution is $f(x) = h\\big(x - q \\floor{x/q}\\big)$, where $h \\colon [0, q) \\to Y$ is arbitrary.\nEvery solution for $p$ is also a solution for each nonzero integer multiple $np$.\nThe analogous equation $f(x + p) = -f(x)$ is called the antiperiodicity equation; its solutions have period $2p$.\n",
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