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. For every sequence $$(a_n)_{n=0}^{\infty}$$ consisting of elements of $$X$$ which converges to $$x_0$$, the sequence $$(f(a_n))_{n=0}^{\infty}$$ converges to $$f(x_0)$$.