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Math Help - Continuity proof

  1. #1
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    Continuity proof

    Suppose that F: U subset of R^n into R^m, with U open. Let x sub zero in the set of the closure of U be fixed. Prove that F is continuous at x sub zero if and only if limit as x aproaches x sub zero of F(x) = F(x sub zero)
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by jburks100 View Post
    Suppose that F: U subset of R^n into R^m, with U open. Let x sub zero in the set of the closure of U be fixed. Prove that F is continuous at x sub zero if and only if limit as x aproaches x sub zero of F(x) = F(x sub zero)
    If \lim_{x\to x_0}f(x)=f(x_0), then \forall~\epsilon>0, \exists~\delta such that |x-x_0|<\delta \implies |f(x)-f(x_0)|<\epsilon.

    This is exactly what it means to be continuous at x_0. In Rudin's "Principles of Mathematical Analysis" he offers this as a second definition of continuity.
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