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Confusion about continuity in metric spaces

  1. Apr 26, 2013 #1
    I'm confused about the the definition of a function not being continuous.

    Is it correct to say f(x) is not continuous at x in the metric space (X,d) if
    [itex]\exists[/itex]ε>0 such that [itex]\forall[/itex][itex]\delta[/itex] there exists a y in X such that d(x,y)<[itex]\delta[/itex] implies d(f(x),f(y))>ε

    Is y dependant on [itex]\delta[/itex]? It seems to me as though it should be.

    An example let f:(ℝ,de)→(ℝ,de) be defined as f(x)=x if x≤0 and f(x)=1+x if x>0 and prove that f(x) is not continuous at x=0.

    So [itex]\exists[/itex]ε=0.5>0 such that [itex]\forall[/itex][itex]\delta[/itex] there exists a y=[itex]\delta[/itex] in ℝ such that d(x,y)<[itex]\delta[/itex] implies d(f(x),f(yδ))>0.5

    Is my understanding and implementation of the defintion correct? To prove the lack of continuity do I have to find a specific y value which is dependant on delta.
     
  2. jcsd
  3. Apr 26, 2013 #2

    Dick

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    Yes, you've got it. Find an ε such that for all δ there is a y value that violates the conditions of continuity.
     
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