Results 1 to 2 of 2

Math Help - Another continuity proof

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    6

    Another continuity proof

    Suppose that F: U subset of R^n into R^m, with U open. If F is Lipshitz continuous at x in the set of U show that F is continuous at x.

    Our teacher said this was trivial, but I of course cannot figure it out.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943
    Quote Originally Posted by jburks100 View Post
    Suppose that F: U subset of R^n into R^m, with U open. If F is Lipshitz continuous at x in the set of U show that F is continuous at x.

    Our teacher said this was trivial, but I of course cannot figure it out.
    For f to be Lipschitz continuous, then \exists~N such that d(f(y),f(x))\leq Nd(y,x).

    Therefore, let \delta=\frac{\epsilon}{2N} and thus

    d(f(y),f(x))\leq Nd(y,x)=\frac{N\epsilon}{2N}=\frac{\epsilon}{2}<\e  psilon

    So f is continuous at x.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Continuity Proof
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 14th 2010, 02:15 AM
  2. continuity proof
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 27th 2010, 01:48 AM
  3. Continuity proof
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: November 9th 2009, 08:34 AM
  4. Continuity proof
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 22nd 2009, 12:41 PM
  5. Continuity proof?
    Posted in the Calculus Forum
    Replies: 2
    Last Post: June 12th 2009, 05:01 AM

Search Tags


/mathhelpforum @mathhelpforum