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 (in symbolic form):
[...]
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.