The Dirichlet function has no limit anywhere

Idea. Near any point, the function keeps jumping between $0$ and $1$ in an extremely dense way. Two sequences, one made of rationals and one of irrationals, will reveal it.

Two sequences. Fix $a\in\mathbb{R}$. For every $n\ge1$, the interval $\left(a,a+\frac1n\right)$ contains a rational number $r_n$ and an irrational number $s_n$. Then $r_n\to a$ and $s_n\to a$.

Values. $D(r_n)=1$ for all $n$, so $D(r_n)\to1$. And $D(s_n)=0$ for all $n$, so $D(s_n)\to0$.

Conclusion. If $D$ had a limit $\ell$ at $a$, both $1$ and $0$ would equal $\ell$. This is impossible, so $D$ has no limit at any point $a$.

Remark. In particular $D$ is not continuous anywhere. The problem "$x$ times the Dirichlet function" shows how multiplying by $x$ changes this.