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 $$x_0$$.
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 $$|x - x_0| < \delta$$