what is the periodicity of $e^{j\omega n}$ when $\omega=0$? what is the periodicity of $e^{j\omega n}$ when $\omega=0$? I think it is $N=1$. But signals and systems book by Oppenheim and Willsky page no.28, mentions it as undefined.  I can understand this for continuous time signals but for discrete time case, I think unconditionally it is $N=1$ because $x(n) = 1$ and so $x(n) = x(n+1)$ for all $n$. What do you think?
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.