types of continuity #maths #realanalysis

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[ro]

The inequality should not be strict for Lipschitz as if x=y, we get 0 < 0.

[xyz.ijk.]

Huge fan, but speeding up the sound like that reduced the utility. [corrected typo for "speeding"]

[MichaelRothwell]

Very nice!

For the examples to work, we need to know what the set A is in each case. Choosing A to be the natural domain in each case (ℝ, ℝ₊, ℝ respectively) works.

But we can be more restrictive:

For the second example, 0 must be included in A (or be an accumulation point), otherwise the gradient of line segments between points on graph is limited and the function is Lipschitz continuous.

For the third example, A must be unbounded, otherwise the gradient of line segments between points on graph is limited and the function is Lipschitz and uniformly continuous.

It curious that having a vertical tangent to the graph (an "infinite gradient") at a single point of a continuous function is less pathological than having an unbounded gradient only when the infinite spread of the domain is considered in the sense that the first maintains uniform continuity but the second doesn't.

[lih]

I've heard it's useful for hamiltonian systems, but does anyone know how?

[rmt]

I don't know the meaning of "continuity" here, but otherwise I mostly follow.