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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.

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The question doesn't say whether this is for freshman calc or for an upper-division class in which students are being reintroduced to concepts such as continuity that they've already seen. In this answer, I'll assume the former.

You need Lipschitz continuity to get uniqueness results for the solutions to certain differential equations. Without it, you can get weirdnesses such as the famous Norton's dome, which, if you take it seriously, means that Newtonian mechanics is nondeterministic. I would have liked it if my freshman calc teacher had told me about these ideas.

However, most of us are teaching students who are almost all completely unlike ourselves at the same age, so it sounds like a bad idea to me to introduce Lipschitz continuity in first-semester freshman calculus. Students at that stage in their education are grappling with several very different things: (1) in most cases, they need to remediate their weak skills in abstract/symbolic/formal reasoning and in arithmetic and basic algebra; (2) they're learning a body of algorithmic techniques for differentiation; and (3) they're learning some very deep and abstract notions such as limits and the completeness property of the reals. At my school (a community college in California), roughly half of each class is destined to fail because they can't handle #1, and of the remainder, very few succeed in #3 (which is why they don't get an A). For this reason, teaching multiple definitions of the limit sounds like a bad idea to me, unless this is something like honors freshman calc at MIT.

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    $\begingroup$ For what it's worth, Estep says his book can be used for either purpose, although he mentions honors in connection with freshman calculus. $\endgroup$ – J W Dec 17 '16 at 19:24

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