Constructing Continuous Functions

摘要

©, The Mathematical Association of America.Summary: This paper introduces a simple geometric construction method for graphs of continuous functions. Graphs of a nowhere differentiable continuous function, a Lipschitz function that is not differentiable on a perfect set, and a continuous function whose level sets are countable are constructed using this method. We argue that introducing students to this method early on in Real Analysis fosters proper geometric intuition about continuity and provides for simple geometric proofs of non-differentiability properties.