# A nowhere continuous function

This is a strange function!

One example is the Dirichlet function $$1_{\mathbb{Q}}$$.
$$1_{\mathbb{Q}}(x)=1$$ if $$x \in \mathbb{Q}$$ and
$$1_{\mathbb{Q}}(x)=0$$ if $$x \in \mathbb{R} \setminus \mathbb{Q}$$.

$$1_{\mathbb{Q}}$$ is everywhere discontinuous because $$\mathbb{Q}$$ is everywhere dense in $$\mathbb{R}$$.

The function $$x \mapsto x \cdot 1_{\mathbb{Q}}(x)$$ is continuous in $$0$$ and discontinuous elsewhere.

## 2 thoughts on “A nowhere continuous function”

1. Jean-Pierre Merx says:

It is a good Mathematical example… the effective computability of the function is indeed another story!

2. Benoît RIVET says:

Is this really a good example ? We are unable to compute the Dirichlet function for many numbers, such as the euler gamma constant or cos(ln(sin(exp(pi))))…