# Continuity question in Topology

1. Apr 22, 2010

### zilla

1. The problem statement, all variables and given/known data
Let (X,d) be a metric space, M a positive number, and f: X->X a continuous function for which:

d(f(x), f(y)) is less than or equal to Md(x,y)

for all x, y in X. Prove that f is continuous. Use this to conclude that every contractive function is continuous.

3. The attempt at a solution

It seems intuitively obvious that f is continuous, but there are a couple of things throwing me off on this one. First, it's asking about concluding that contractive functions are continuous, but this isn't by definition a contractive function that we're looking at since M >1 is a possibility. And second, the problem 'gives' that f is continuous and then asks you to prove that f is continuous. Help!

2. Apr 22, 2010

### VeeEight

I think the problem is asking you to show that Lipschitz maps are continuous. Contractions are a special case of Lipschitz maps, so the result follows when you show the first propety. You are not supposed to assume f is continuous.

Use the epsilon delta definition of continuity. Write out the inequality with M. From this, can you choose a delta, perhaps involving M, to satisfy the continuity condition? After you have tried to find this delta, consider uniform continuity.

Last edited: Apr 22, 2010
3. Apr 22, 2010

### zilla

Could you elaborate on what you mean by the 'first property'?

Also, so I'm looking for |f(x) - f(y)| < Epsilon?

4. Apr 22, 2010

### VeeEight

5. Apr 22, 2010

### zilla

Thanks for the help - I appreciate it.