1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Continuity question in Topology

  1. Apr 22, 2010 #1
    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. jcsd
  3. Apr 22, 2010 #2
    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
  4. Apr 22, 2010 #3
    Could you elaborate on what you mean by the 'first property'?

    Also, so I'm looking for |f(x) - f(y)| < Epsilon?
  5. Apr 22, 2010 #4
  6. Apr 22, 2010 #5
    Thanks for the help - I appreciate it.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook