Results 1 to 2 of 2

Math Help - Prove a convex function is continuous

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    195

    Prove a convex function is continuous

    I need to prove that a real valued convex function on R is continuous. I'm thinking this is only a special case and may not be the right way to attack the problem:

    Assume the contrary. That is, there is at least one point x_0 such that for any \epsilon > 0 there is no \delta > 0 such that d(x,x_0)<\delta implies |f(x_0) - f(x)| < \epsilon.

    Now, if we further assume that there exists x,y such that x_0\in (x,y) and d(f(x),f(y))<\epsilon, then it follows that since x_0 = \lambda x + (1-\lambda) y for some \lambda \in (0,1) we have

    |f(x_0) - f(x)| = |f(\lambda x + (1-\lambda) y) - f(x)| \le (1-\lambda)|f(y)-f(x)| \le \epsilon

    This seems like an elegant application of convexity, but it only applies to simple discontinuities, and I'm not sure I can extend it to the more general case.

    Thanks!! edit: not sure whats wrong with my latex :/
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: Prove a convex function is continuous

    See theorem 3.2 here W.rudin, Real and Complex Analysis . Ask your doubts.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. prove that every continuous function is separately continuous
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: November 23rd 2011, 03:57 AM
  2. Prove A Function Is Convex?
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: May 1st 2011, 01:48 PM
  3. Replies: 7
    Last Post: May 9th 2010, 05:29 AM
  4. prove function is continuous
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: April 9th 2010, 10:49 AM
  5. Prove f is a continuous function
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: March 1st 2010, 12:18 PM

Search Tags


/mathhelpforum @mathhelpforum