For a function $f:\mathbb{R}\rightarrow \mathbb{R}$ denote $f_0(x)=x$, $f_n(x)=f(f_{n-1}(x))$. Assume that $f$ satisfies a functional equation $$f_n(x)+a_{n-1} f_{n-1}(x)+\dots+a_0 f_0(x)\equiv 0$$ for some constant real coefficients $a_{0},a_1,\dots,a_{n-1}$. Assume also that $f$ is continuous. What are conditions for polynomial $t^n+a_{n-1}t^{n-1}+\dots +a_0$ which allow to conclude that $f$ is linear?

existenceof solutions are discussed. THere are solutions if and only if the corresponding polynomial has a real root. $\endgroup$of the same sign, then there are non-linear piecewise-linear solutions. Say, for $t^2-3t+2$, a function $f(x)=(3x+|x|)/2$ fits. $\endgroup$