We say that a function f : A R is Lipschitz on A if there...

 

Transcribed Image Text:

We say that a function f : A → R is Lipschitz on A if there is some fixed k> 0 such that for all x, y € A, we have |f(x) = f(y)| ≤ k|xy|. (5 points) Let A be connected. Suppose f : A → R is differentiable. Prove that if f has bounded derivative on A, then f is Lipschitz on A. Indeed, the converse is also true but this is not necessary for this problem. (5 points) Prove that if f is Lipschitz on A, then f is uniformly continuous on A. We say that a function f : A → R is Lipschitz on A if there is some fixed k> 0 such that for all x, y € A, we have |f(x) = f(y)| ≤ k|xy|. (5 points) Let A be connected. Suppose f : A → R is differentiable. Prove that if f has bounded derivative on A, then f is Lipschitz on A. Indeed, the converse is also true but this is not necessary for this problem. (5 points) Prove that if f is Lipschitz on A, then f is uniformly continuous on A.

Answer rating: 100% (QA)

Proof 1 Suppose A is a connected set and f A R is differentiable If f has a bounded derivative on A ... View the full answer