Školjka

%V0 Let $f: \mathbb{R}\to\mathbb{N}$ be a function which satisfies $f\left(x + \dfrac{1}{f(y)}\right) = f\left(y + \dfrac{1}{f(x)}\right)$ for all $x$, $y\in\mathbb{R}$. Prove that there is a positive integer which is not a value of $f$. Proposed by Žymantas Darbėnas (Zymantas Darbenas), Lithania