The period of the function $\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}1, \mathrm{when} \mathrm{x} \mathrm{is} \mathrm{a} \mathrm{rational} \\ 0, \mathrm{when} \mathrm{x} \mathrm{is} \mathrm{irrational} \end{array}\right\} \mathrm{is}$

For every rational number T, we have
$\mathrm{f}(\mathrm{T}+\mathrm{x})=\left\{\begin{array}{l}1,\text{ when }\mathrm{x}\text{ is a rational }\\ 0,\text{ when }\mathrm{x}\text{ is irrational }\end{array}\right\}=\mathrm{f}\left(\mathrm{x}\right)$ 
but there is no least positive value of T for which f (T + x) = f (x) because there are infinite number of rational numbers between any two rational numbers. Therefore, f (x) is a periodic function having no fundamental period.