1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?