[Lecture 10] Sequential Characterization of Continuity

Suppose \(E\) is a non-empty subset of \(\mathbb{R}\). Let \(f: E \rightarrow \mathbb{R}\) and \(a \in E\). Then the following statements are equivalent

1. \(f\) is continuous at \(a \in E\).
2. If \(x_n\) converges to \(a\) and \(x_n \in E\), then \(f(x_n) \rightarrow f(a)\) as \(n \rightarrow \infty\).