Results 1 to 2 of 2

Math Help - Proof of continuity in two dimensions

  1. #1
    Member
    Joined
    Dec 2010
    Posts
    107

    Proof of continuity in two dimensions

    Firstly, i'm sorry if this is the wrong forum. It seems to be analysis, but it is in a calculus book so wasn't sure.
    I'm stuck on this question:
    Prove that if  g:\mathbb{R} \rightarrow \mathbb{R} is continuous at a then f(x,y)=g(x) is continuous at (a,b) for all b \in \mathbb{R}

    If someone could just give me an idea of how to start the proof. I understand the definition for continuity.
    Last edited by worc3247; April 25th 2011 at 12:20 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member HappyJoe's Avatar
    Joined
    Sep 2010
    From
    Denmark
    Posts
    234
    Quote Originally Posted by worc3247 View Post
    \mapsto Firstly, i'm sorry if this is the wrong forum. It seems to be analysis, but it is in a calculus book so wasn't sure.
    I'm stuck on this question:
    Prove that if g:\mathbb{R} \rightarrow \mathbb{R} is continuous at a then f(x,y)=g(x) is continuous at (a,b) for all b \in \mathbb{R}

    If someone could just give me an idea of how to start the proof. I understand the definition for continuity.
    So, you are given that g is continuous. This means that for any epsilon>0, there exists a delta>0, such that |a-x|<delta implies |g(a)-g(x)|<epsilon. You want to prove a similar implication for f, that is, for a given epsilon>0, you need the existence of a delta>0, such that |a-x|<delta implies |f(a,b)-f(x,b)|<epsilon (where b is arbitrary).

    However, by definition of f, we have that |f(a,b)-f(x,b)| = |g(a)-g(x)|. Let me know, if you can see where this goes.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proof about nullspace dimensions of linear operators
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: October 21st 2010, 08:00 PM
  2. Continuity Proof
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 2nd 2010, 06:05 PM
  3. Continuity Proof
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 14th 2010, 02:15 AM
  4. Proof of Midpoint and Vectors in 3 Dimensions
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 21st 2009, 12:24 PM
  5. proof of continuity
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 16th 2007, 12:26 PM

Search Tags


/mathhelpforum @mathhelpforum