Let $f : \mathbb R \rightarrow \mathbb R$ be a continuous function such that $f(f(x)) + f(x) + x = 0$ for all $x \in \mathbb R$. Find all $f(x)$.

Question

Let $f : \mathbb R \rightarrow \mathbb R$ be a continuous function such that $f(f(x)) + f(x) + x = 0$ for all $x \in \mathbb R$. Find all $f(x)$.

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I already proved that $f(x)$ is injective. But I didn’t know how to use the fact that $f$ is continuous. Can anyone give a hint or a solution to this problem? Thank you.