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  "body": "\n## Definition\n\nThe *Pexider equation* is the [[functional equation|SRAN4EKWS0]]\n$$f(x + y) = g(x) + h(y) \\quad \\text{for all } x, y \\in \\R,$$\nfor three unknown [[functions|0EPAT51DXQ]] $f, g, h \\colon \\R \\to \\R$.\nA solution is a triple $(f, g, h)$ that satisfies this identity for all real $x$ and $y$.\n\n## Remarks\n\nExample: $f(x) = 5x + 3$, $g(x) = 5x + 1$, $h(x) = 5x + 2$ is a solution, since both sides equal $5x + 5y + 3$.\nNon-example: $f(x) = x^2$, $g(x) = x^2$, $h(x) = x^2$ is not a solution, since $f(1 + 1) = 4$ while $g(1) + h(1) = 2$.\nSetting $y = 0$ and then $x = 0$ gives $g(x) = f(x) - h(0)$ and $h(y) = f(y) - g(0)$.\nThen $A(x) = f(x) - g(0) - h(0)$ is an [[additive function|KF719SZD5T]], which gives the general solution $f = A + a + b$, $g = A + a$, $h = A + b$ with $a = g(0)$ and $b = h(0)$.\nThe same method solves the equation for maps from a group to an abelian group.\n",
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