Question 10
Let $f: \mathbb{R} \rightarrow \mathbb{N}$ be a function which satisfies
$$
f\left(x+\frac{1}{f(y)}\right)=f\left(y+\frac{1}{f(x)}\right) \quad \text{ for all } x, y \in \mathbb{R}
$$
Prove that there is a positive integer which is not a value of $f$.
$$
f\left(x+\frac{1}{f(y)}\right)=f\left(y+\frac{1}{f(x)}\right) \quad \text{ for all } x, y \in \mathbb{R}
$$
Prove that there is a positive integer which is not a value of $f$.