Math and science::Analysis::Tao::09. Continuous functions on R
Equivalent formulations of function continuity
Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \)
be a function, let \( x_0 \) be an element of \( X \). Then the following
four statements are logically equivalent:
\( f \) is continuous at \( x_0 \).
[...]
For any real \( \varepsilon > 0 \) there exists a real \( \delta > 0 \) such
that \( |f(x) - f(x_0)| < \varepsilon \) for all \(x \in X \) and
\( |x - x_0| < \delta \)
