ISL 2008 A6

Let $f: \mathbb{R} \rightarrow \mathbb{N}$ be a function which satisfies $$f\left(x+\frac{1}{f(y)}\right)=f\left(y+\frac{1}{f(x)}\right) \text { for all } x, y \in \mathbb{R} .$$ Prove that there is a positive integer which is not a value of $f$.