SCHWEITZER 1968 Problem 4

Let $f$ be a complex-valued, completely multiplicative,arithmetical function. Assume that there exists an infinite increasing sequence $N_k$ of natural numbers such that $$f(n)=A_k \not= 0 \;\textrm{provided}\ N_k \leq n \leq N_k+4 \sqrt{N_k}\ .$$ Prove that $f$ is identically $1$ .

I. Katai