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  "body": "\n## Summary\n\nJensen's functional equation is the [[functional equation|SRAN4EKWS0]] $f\\left(\\frac{x + y}{2}\\right) = \\frac{f(x) + f(y)}{2}$ for an unknown function $f \\colon \\R \\to \\R$.\nIt says that the value at the midpoint of two points is the average of the values.\nIts solutions are the functions $f = A + b$ with $A$ an [[additive function|KF719SZD5T]] and $b$ a constant, and the continuous ones are the affine functions $f(x) = ax + b$.\nIt is the equality case of midpoint convexity, which links it to [[convex functions|07Q7CNM5WF]].\n\n## Explanations\n\n### L1 human\n\nTake two points on the graph and their midpoint on the $x$-axis.\nThe equation asks the graph to pass exactly through the middle of the chord between the two points.\nA straight line does this: for $f(x) = 2x + 3$, the midpoint of $0$ and $2$ is $1$, and $f(1) = 5$ is the average of $f(0) = 3$ and $f(2) = 7$.\nA curved graph such as $x^2$ fails, since it sags below its chords.\n",
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