Question 4
Let $\mathbb{Z}_{\geq 0}$ be the set of all nonnegative integers. Let $f: \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0}$ be an unbounded function such that, if $m$ and $n$ are nonnegative integers satisfying
$$ f(m+n)=\max\left\{f(0),f(1),\cdots,f(m+n)\right\}, $$
then
$$ f(m+n)=f(m)+f(n). $$
Prove that there exist positive integers A, B, C and D such that for all nonnegative integers n, $ f(An + B) = Cn + D $.
We say that f is unbounded if for each nonnegative integer N, there exists some nonnegative integer n such that $ f(n) \geq N $.
$$ f(m+n)=\max\left\{f(0),f(1),\cdots,f(m+n)\right\}, $$
then
$$ f(m+n)=f(m)+f(n). $$
Prove that there exist positive integers A, B, C and D such that for all nonnegative integers n, $ f(An + B) = Cn + D $.
We say that f is unbounded if for each nonnegative integer N, there exists some nonnegative integer n such that $ f(n) \geq N $.