Fundamental Period vs function continuous atleast once

Question

I know that function is periodic, non constant and continuous at least once $\implies$ function has a fundamental period

My question:

Can we construct a function which is discontinuous everywhere and has fundamental period ?

I constructed following example:

$f(x)=\begin{cases} 4,&x\in \mathbb{Q}\\\{x\},&x\in\mathbb R-\mathbb Q\end{cases}$

Its fundamental period is 1