ABSTRACT
This chapter proves the notion that a continuous function is a function whose graph has no holes, jumps, or asymptotes. It establishes the basic theorems about continuous functions and explores sequential continuity graphically and numerically. Although graphs and numerical calculations help to explore functions to make conjectures about their continuity, they do not give proofs of those conjectures. Proofs come from using the definition of continuity. The chapter gives several proofs of the continuity of specific functions and uses the limit laws to prove that certain combinations of continuous functions are continuous. It discusses the intermediate value theorem (IVT) and develops a root-finding algorithm from the proof of the IVT. The chapter also includes exercise problems related to the continuous functions.
