\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \)

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:

1. \( f \) is continuous at \( x_0 \).
2. [...]
3. 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 \( |x - x_0| < \delta \)
4. For any real \( \varepsilon > 0 \) there exists a real \( \delta > 0 \) such that \( |f(x) - f(x_0)| \le \varepsilon \) for all \(x \in X \) and \( |x - x_0| \le \delta \)