Question

Does there exist a function which is periodic but has no fundamental period?

Answer 1

$f\left(x\right)=\{\begin{array}{c}0;x\text{is irrational}\\ 1;x\text{is rational}\end{array}$

$f(x+T)=\{\begin{array}{c}0;x\text{is irrational}\\ 1;x\text{is rational}\end{array}$

Hence, $f(x+T)=f\left(x\right)$

$f\left(x\right)$ is periodic function but has no fundamental period.