# Lipschitz continuity before standard definition of continuity

In Practical Analysis in One Variable, Donald Estep introduces Lipschitz continuity early on, delaying the standard definition of continuity, along with uniform continuity, until the beginning of his more theoretical section on analysis in the final part of the book. According to Estep, Lipschitz continuity makes it easier to give constructive proofs of several key results.

Lipschitz continuity captures the idea of making the change in output arbitrarily small by making the change in input small. A nonnegative Lipschitz constant, $L$, if it exists, provides an upper bound for the change in output based on the change in input: $|f(x_1)-f(x_2)|\leq L|x_1-x_2|$. That said, $x^{1/3}$ for instance fails to be Lipschitz continuous on any interval containing zero, although it is continuous by the standard definition.

What are the pros and cons of early Lipschitz continuity? I can imagine that its quantitative approach of calculating an upper bound might be easier for some students to grasp than using epsilon-delta. If you have any experience with using it, I would be interested to know.