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## Constant function

Let $g: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $\lim_{x\to \infty}g(x)-x=\infty$ and such that the set $\{x : g(x)=x\}$ is finite and nonempty. Prove that if $f: \mathbb{R} \to \mathbb{R}$ is continuous and $f\circ g=f$, then $f$ is constant.
AMM 10818