If $f$ is a monotonically increasing function and if $$f\left({x+f(x)\over2}\right)=x$$ for every $x\in\mathbb R$, prove that $$f(x)=x$$
2025-01-13 00:13:23.1736727203
Monotonically increasing functions 123
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Suppose $f(x)\ne x$ for some $x .$ Let $ (x+f(x))/2 =y.$ CASE 1. $f(x)>x .$ Then $ y>x$ but $ f(y)=x<f(x) $, a contradiction.CASE 2. $f(x)<x. $ Then $y<x$ but $ f(y)=x>f(x) $, a contradiction.