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  "body": "\n## Definition\n\n*Jensen's functional equation* is the [[functional equation|SRAN4EKWS0]]\n$$f\\left(\\frac{x + y}{2}\\right) = \\frac{f(x) + f(y)}{2} \\quad \\text{for all } x, y \\in \\R,$$\nfor an unknown [[function|0EPAT51DXQ]] $f \\colon \\R \\to \\R$.\nA solution is called a *Jensen function*.\n\n## Remarks\n\nExample: $f(x) = 2x + 3$ is a solution, since both sides equal $x + y + 3$.\nNon-example: $f(x) = x^2$ is not a solution, since the right side minus the left side is $(x - y)^2 / 4$, which is positive for $x \\ne y$.\nIf $f$ is a solution, then $A(x) = f(x) - f(0)$ satisfies $A(x/2) = A(x)/2$ and then $A(x + y) = A(x) + A(y)$.\nSo the solutions are $f = A + b$ with $A$ an [[additive function|KF719SZD5T]] and $b \\in \\R$.\nThe same equation makes sense for functions on a real interval or on a convex subset of a vector space.\n",
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