What do you want explained about it? There are several "Dirichlet" functions but the standard one is:

f(x)= 0 if x is rational, 1 if x is irrational.

That has the property that it is discontinuous for all x because in any interval around x, there exist both rational and irrational numbers, so values of 0 and 1 arbitrarily close to x. What is it that is confusing you?