Continuity Definitions

Let \(E\) be a non-empty set of \(\mathbb{R}\). Let \(f: E \rightarrow \mathbb{R}\). Then

1. \(f\) is said to be continuous at a point \(a \in E\) if and only if given \(\epsilon > 0\), there is a \(\delta > 0\) (which depends on \(\epsilon, f, a\) such that $$ 0 < |x - a| < \delta \quad \text{ and } \quad x \in E \quad \text{ implies } |f(x) - f(a)| < \epsilon. $$
2. \(f\) is said to be continuous on \(E\) (notation \(f: E \rightarrow \mathbb{R}\) is continuous) if and only if \(f\) is continuous at every \(x \in E\).

Let \(E\) is a non-empty subset of \(\mathbb{R}\). A function \(f: E \rightarrow \mathbb{R}\) is said to be bounded on \(E\) if and only if there is an \(M \in \mathbb{R}\) such that \(|f(x) \leq M\) for all \(x \in E\), which case we shall say that \(f\) is dominated by \(M\) on \(E\).