Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Uniform continuity

  1. Jan 28, 2009 #1
    This may sound lame but I am not able to get the definition of uniform continuous functions past my head.

    by definition:
    A function f with domain D is called uniformly continuous on the domain D if for any eta > 0 there exists a delta > 0 such that: if s, t D and | s - t | < delta then | f(s) - f(t) | < eta. Click here for a graphical explanation.

    I can just choose "delta" that is a large number that will make any 2 points on the curve satisfy this condition. 1/x would be uniform continuous if I simply choose a large enough delta.

    moreover, what is the utility and application of uniform continuous fynctions?

  2. jcsd
  3. Jan 28, 2009 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I think what you're missing is that in the definition, the part that says

    "if s, t D and | s - t | < delta then | f(s) - f(t) | < eta"

    actually means

    "if for all s,t in D such that |s-t|<delta, we have | f(s) - f(t) | < eta."

    So it does not suffice that you can find two points of D a distance less than delta apart that satisfy | f(s) - f(t) | < eta, but rather, the definition is saying that all the points in D that are a distance less than delta apart must satisfy | f(s) - f(t) | < eta !
  4. Jan 28, 2009 #3


    User Avatar
    Gold Member

    For starters, in order to prove that the Darboux integral is defined for any continuous function on a closed interval, Cantors theorem - which states that a continuous function on a closed interval is uniformly continuous - is used and then the uniform continuity is used to prove that the integral is defined.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Uniform continuity
  1. Uniform Continuity (Replies: 14)

  2. Uniform Continuity (Replies: 5)

  3. Uniform continuity (Replies: 2)