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Homework Help: Quick Continuity question

  1. Apr 9, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove that if g:R->R is continuous at a then f(x,y)=g(x) is continuous at (a,b) [tex]\forall[/tex] b [tex]\in[/tex] R

    2. Relevant equations

    3. The attempt at a solution

    So we know
    [tex]\forall[/tex]e>0 [tex]\exists[/tex]d>0 s.t. [tex]\forall[/tex]x[tex]\in[/tex]R where |x-a|<d we have |g(x) - g(a)|<e
    So I've said as [tex]\forall[/tex]b[tex]\in[/tex]R g(x)=f(x,y) & g(a)=f(a,b), these can be substituted in giving the expression we need except for the condition that [(x-a)2 + (y-b)2]1/2<d.
    This seems to be an incorrect cheat though, am I along the right lines or not?
  2. jcsd
  3. Apr 9, 2010 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    You are looking at the right line of thought.

    If |(x,y)-(a,b)|<d, what can you say about |x-a|?
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