RMM 2020 Problem 4

Let $\mathbb N$ be the set of all positive integers. A subset $A$ of $\mathbb N$ is sum-free if, whenever $x$ and $y$ are (not necessarily distinct) members of $A$ , their sum $x+y$ does not belong to $A$ . Determine all surjective functions $f:\mathbb N\to\mathbb N$ such that, for each sum-free subset $A$ of $\mathbb N$ , the image $\{f(a):a\in A\}$ is also sum-free.

Note: a function $f:\mathbb N\to\mathbb N$ is surjective if, for every positive integer $n$ , there exists a positive integer $m$ such that $f(m)=n$ .