Herr Lipschitz: Continuously Yours!

Basic calculus: Continuity: Let A ⊂ R and f : A → R be continuous. Then for each x0 ∈ A and for given ε > 0, there exists a δ(ε, x0) > 0 such that x ∈ A and | x − x0 |< δ imply | f(x) − f(x0) |< ε. (Observe that δ depends, in general, on ε as well as the point x0).

A little more advanced: Uniform Continuity (Definition): A function f : A → R, where A ⊂ R is said to be uniformly continuous on A if given ε > 0, there exists δ > 0 such that whenever x, y ∈ A and | x − y |< δ, we have | f(x) − f(y) |< ε.

Question 1: What is the difference between continuity and uniform continuity? Which is the stronger condition? Why?

Once you've clearly resolved the above and looked at a few examples to see the difference clearly (the internet is a great resource if you know where to look!):

Question 2 (Self Study): What is Lipschitz Continuity? Is it a stronger or weaker condition that continuity and uniform continuity?

After you've resolved the above at the level of definition and theory, make sure you look at few examples and solve a few problems so that the concepts settle in your mind clearly.

PS: Even if you're not a mathematician, if you're using functions, differential equations, etc., you ought to be curious about what you're using and why it works!

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