Results 1 to 2 of 2

Math Help - Continuous function on a compact subset of R

  1. #1
    Member
    Joined
    Feb 2009
    Posts
    103

    Continuous function on a compact subset of R

    Here's the question:

    "Let K be a compact subset in \Re and let f : K \rightarrow \Re be a continuous function. Prove that for every \epsilon > 0 there exists L_{\epsilon} such that

    |f(x) - f(y)| \leq L_{\epsilon} |x-y| + \epsilon for every x,y \in K."

    I don't really see the "point" of the question - can't we just take L_{\epsilon} = |f(x) - f(y)| / |x-y| ? What are we "supposed" to do?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Amanda1990 View Post
    Here's the question:

    "Let K be a compact subset in \Re and let f : K \rightarrow \Re be a continuous function. Prove that for every \epsilon > 0 there exists L_{\epsilon} such that

    |f(x) - f(y)| \leq L_{\epsilon} |x-y| + \epsilon for every x,y \in K."

    I don't really see the "point" of the question - can't we just take L_{\epsilon} = |f(x) - f(y)| / |x-y| ? What are we "supposed" to do?
    You can't just take L_{\epsilon} = |f(x) - f(y)| / |x-y| , because L_{\epsilon} has to be a constant. Suppose for example that f is the function f(x) = \sqrt x defined on the interval K=[0,1]. Then |f(x) - f(y)| / |x-y| becomes unbounded when x=0 and y→0.

    I think the trick is to consider two separate cases: when x and y are close together, and when they are not.

    Given \epsilon>0, the uniform continuity of f tells you that there exists a \delta>0 such that |f(x) - f(y)| < \epsilon whenever |x-y|<\delta. That deals with the case when x and y are close together (in other words, within δ of each other). Now you just have to show that |f(x) - f(y)| / |x-y| is bounded when |x-y|\geqslant\delta. (I'll leave that part to you.)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Closed subset of sequentially compact set is sequentially compact.
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: September 13th 2011, 02:25 PM
  2. compact space and continuous function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 23rd 2010, 01:53 PM
  3. inf and sup of a compact subset of R
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: February 28th 2010, 09:04 PM
  4. compact subset of R
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 3rd 2009, 03:56 PM
  5. Compact subset problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 19th 2007, 04:22 PM

Search Tags


/mathhelpforum @mathhelpforum