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Homework Help: Continuity on an interval

  1. Feb 2, 2009 #1
    1. The problem statement, all variables and given/known data

    Suppose a function is continuous at a point, c. Does this mean there exists an interval around c which is also continuous?

    If so prove

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 2, 2009 #2

    Dick

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    Any opinion on whether it might be true or not? Doesn't seem true to me, but that's just an opinion also because I can't think why it would be. You might try to find a counterexample first.
     
  4. Feb 5, 2009 #3
    it boils down to the definition of the limit.

    for all e>0. there exists s>0 such that

    if x satisfies abs(x-a) then abs(f(x)-a)<e

    the question is: Does f have to be defined on the interval abs(x-a)?

    example of this- A function is undefined at every point except a.

    does the limit exist at a?

    if yes, then we have a trivial counterexample to the original post
     
  5. Feb 5, 2009 #4

    Dick

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    The counterexample isn't that trivial. Define f(x)=x if x is rational and f(x)=0 if x is irrational. Where is that continuous?
     
  6. Feb 5, 2009 #5
    ah, thanks for the counterexample.it would only be continuous at 0.


    My second post was to clarify a technical point.
    If f is undefined at every point except a, and defined at a, is f continuous at a?
     
  7. Feb 5, 2009 #6

    Dick

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    Continuity says as x->a, f(x)->f(a). If there are undefined points arbitrarily close to a, I would say no, it's not continuous. If you say the definition is x->a AND f(x) defined at x, then you could say yes, it is. A 'function' with 'undefined' points is a little ambiguous. In any event, even you decide to call it technically continuous, it's not a very interesting example, is it?
     
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