what is the periodicity of $e^{j\omega n}$ when $\omega=0$?
Mathematically speaking, the constant function is periodic with any period $P&gt;0$, because is satisfies $f(x)=f(x+P)$ for all $x$.

The existence of a fundamental period depends on whether $x$ is discrete or continuous. In the first case the smallest positive period is $P=1$. In the second case there is no smallest period and therefore the function does not have a fundamental period.

In any case, claiming it is "undefined" is not accurate. This is all very well defined.