Suppose $X$ is a normal (Hausdorff) zero dimensional topological space and $C\subset X$ is a closed set. Let $f:C\rightarrow \mathbb N$ be continuous. By the Tietze extension theorem, there exists a continuous $g:X\rightarrow \mathbb R$ such that $f\subset g$. I want to know if we can guarantee that there exists a continuous $g:X\rightarrow \mathbb N$ such that $f\subset g$ (even if f is not bounded).
I know that in general it's not true since that would imply that every normal topological space that has at least two points is not connected, therefore I added the zero dimensional hypothesis.
Edit: If it's not true, can I fix it by adding an extra hypothesis?