Math and science::Analysis::Tao::09. Continuous functions on R

# Continous functions

Let $$X$$ be a subset of $$\mathbb{R}$$, and let $$f: X \to \mathbb{R}$$ be a function. Let $$x_0$$ be an element of $$X$$. We say that $$f$$ is continuous at $$x_0$$ iff we have:

[$\text{some limit = what?}$]

in other words, the limit of $$f(x)$$ as $$x$$ converges to $$x_0$$ in X [...].

We say that $$f$$ is continuous on $$X$$ (or simply continuous) iff $$f$$ is continuous at $$x_0$$ for every $$x_0 \in X$$. We say that $$f$$ is discontinous at $$x_0$$ iff it is not continous at $$x_0$$.

Tao describes this definition as one of the most fundamental notions in the theory of functions.

Question: what is Dirichlet's function and what is Thomae's function? How do they relate to function continuity?