Dirichlet's function

Dirichlet's function $f:\mathbb{R}\to\mathbb{R}$ is defined as

This function has the property that it is continuous at every irrational number and discontinuous at every rational one.

Another function that often goes by the same name is the function

This nowhere-continuous function has the surprising analytic expression
This is often given as the (amazing!) example of a sequence of everywhere-continuous functions whose limit function is nowhere continuous.