Continuity question in Topology

[zilla]

Homework Statement

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.

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!

 

[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.

 

[zilla]

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.

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

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

 

[VeeEight]

By "the result follows when you show the first property", I mean that when you show that Lipschitz maps are continuous, you will have shown that contractions are continuous.

Yes, that is what you are looking for. See the definition of continuity: http://en.wikipedia.org/wiki/Continuity_(topology)#Continuous_functions_between_metric_spaces
This is a typical epsilon delta type proof.

 

[zilla]

Thanks for the help - I appreciate it.