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Can a function be continuous on a composed interval?

  1. Jan 27, 2010 #1
    Can a function be continuous on a composed interval? For example, if [tex]f(x)=\frac{1}{x} [/tex] then on the interval [tex] (-\infty,0) \cup (0,\infty), f(x)[/tex] is continous? Or is the function [tex]f(x)[/tex] continuous on [tex] (-\infty,0) [/tex] by itself and [tex] (0,\infty)[/tex] by itself (If you don't get what I'm trying to say reply back)?
     
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  3. Jan 27, 2010 #2

    mathman

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    Re: Continuity

    If you study the general definition of continuity, using topological spaces, there is no requirement that the domain be a connected set. So in general, the answer to your question is yes.
     
  4. Jan 27, 2010 #3

    Landau

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    Re: Continuity

    It's the same. Continuity is a local concept: a function is continuous on some domain D if it is continuous at every point in D. Since your f is continuous at every point in [tex]\mathbb{R}-\{0\}[/tex], it is continuous on every subset [tex]D\subseteq\mathbb{R}-\{0\}[/tex], in particular on [tex]D= (-\infty,0) \cup (0,\infty)[/tex].
     
  5. Jan 28, 2010 #4
    Re: Continuity

    Great question with great answers!
     
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