Uniformly Continuous means 23 Exercise Prove that if D cR an

Uniformly Continuous means:

2.3. Exercise. Prove that if D cR and f: D R is Lipschitz, then f is uniformly continuous.

Solution

Answer:

Suppose D R and f: DR is Lipschitz then by definition of Lipschitz fuction there exists k>0 such that

| f(x) - f(y) | k | x - y | x , y D then f is said to be a Lipschitz function (or to satisfy a Lipschitz condition)

on D.

Now we shall prove that f is uniformly continuous.

Let >0 choose = / k , then for all x ; y E Dwith | x - y | < we have | f(x) - f(y) | < which shows that f is uniformly continuous on E.