KöMaL 2024 B5393

Let $f$ be a real function (that is, a real-valued function of a real variable) satisfying $\left|f(x+y+z)+\sin x+\sin y+\sin z \right|\le 3$, for all $x$, $y$, $z\in \mathbb{R}$. Prove that $|f(x)-\sin x|\le 1$ for all $x\in \mathbb{R}$.