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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?