Continuous Functions Definitions

A function \( f: A \rightarrow \mathbb{R} \) is continuous at a point \( c \in A \) if, for all \( \epsilon > 0 \), there exists a \(\delta > 0\) such that whenever \( |x - c| < \delta \) (and \( x \in A \)) it follows that \[ |f(x) - f(c)| < \epsilon \]