The period of the function f(x) ={1,    when  x  is  a  rational0,    when  x  is  an  irrational is:

For every rational number T, we have                  f(T+x) ={1,    when  x  is  a  rational0,    when  x  is  an  irrational 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.