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Continuity proof, not sure how to put it together.

  1. Feb 28, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove that if f is continuous at a, then for any ε>0 there is a σ>0,?
    such that if abs(x-a)< σ and abs(y-a)< σ then abs[f(x) - f(y)]< ε

    2. Relevant equations

    Definition of continuity and triangle inequality
    abs(f(x)-f(y))= abs(f(x)-f(a) + f(a)-f(y))≤ abs(f(x)-f(a))+ abs(f(y)-f(a))

    3. The attempt at a solution

    So i think I need to apply the continuity definition twice and bring things together with the triangle inequality but i don't know how to go about it all.
     
  2. jcsd
  3. Feb 28, 2012 #2

    HallsofIvy

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    Yes, that's exactly what you need. Since f is continuous at a, given any [itex]\epsilon> 0[/itex] there exist [itex]\delta_1> 0[/itex] such that if [itex]|x- a|< \delta_1[/itex] then [itex]|f(x)- f(a)|< \epsilon/2[/itex] and there exist [itex]\delta_2> 0[/itex] such that if [itex]|y- a|< \delta_2[/itex] then [itex]|f(y)- f(a)|< \epsilon/2[/itex].

    Take [itex]\delta= min(\delta_1, \delta_2)[/itex] so that if [itex]|x- a|< \delta[/itex], both are true.
     
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