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Homework Help: Sequence proof for discontinuity

  1. Mar 2, 2008 #1
    [SOLVED] Sequence proof for discontinuity

    1. The problem statement, all variables and given/known data
    Given a function h(x) = x for all rational numbers x and h(x) = 0 for all irrational numbers, prove that h(x) is continuous at the point x=0 and nowhere else.


    2. Relevant equations
    A function is continuous at a point x0 if and only if for all sequences xn converging to x0, f(xn) converges to f(x0). Therefore, a function is discontinuous if there exists a sequence xn converging to x0 that does not cause f(xn) to converge to f(x0).

    Also, because either a sequence proof or an epsilon-delta proof works here:

    A function is continuous at a point x0 in dom(f) if |x-x0| < delta implies |f(x) - f(x0)| < epsilon


    3. The attempt at a solution

    I'm having a hard time even figuring out where to begin on this. The Denseness of Q theorem states that for every a < b, where a and b are real numbers, there exists a rational number r such that a < r < b. The same applies for irrationals. Therefore, h(x) looks like two different functions, f(x) = 0 and g(x) = x, since Q and not Q are so densely populated. This means that it appears that there is an intersection between these two functions at x = 0.

    For a sequence proof, I really don't have a grasp of how sequence proofs for discontinuity (or continuity, for that matter) work. I guess if you let xn = x and took the limit as x->0, that implies that the limit as h(x) approaches 0 is 0 just by looking at the graph, but I don't think that's analytical enough.

    As far as an epsilon-delta proof goes, I'm not sure what to use as f(x) [or h(x) in this case]. I figure that you could say |x - 0| < delta implies |x - 0| < epsilon, meaning if you let delta = epsilon, |x| < delta implies |x| < epsilon. Is that sufficient?

    That being said, there's also the part where I have to prove that it's not continuous for any other point in dom(h), but it's not defined enough to do that with epsilon-delta, and again, I don't understand the sequence definition enough to apply it.

    Thanks for your help!
     
  2. jcsd
  3. Mar 2, 2008 #2

    HallsofIvy

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    Science Advisor

    Yes, that is basically what you need to do. For a not equal to 0, take a sequence of rational numbers, ri, converging to a. Because those are rational numbers, f(ri)= ri and those converge to what? Take a sequence of irrational numbers, si, converging to a. Because those are irrational numbers, f(si)= 0. What does that sequence converge to? For what a are those the same?

     
    Last edited by a moderator: Mar 2, 2008
  4. Mar 2, 2008 #3
    Great, thanks!
     
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