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Real Analysis - Continuity

  1. Mar 26, 2008 #1
    Two problems, actually, but they are very similar. Here goes:

    1. The problem statement, all variables and given/known data
    Let f be a continuous real-valued function with domain (a, b). Show that if f(r) = 0 for each rational number r in (a, b,), then f(x) = 0 for all x in (a, b).

    2. Relevant equations

    3. The attempt at a solution


    1. The problem statement, all variables and given/known data
    Let f and g be continuous real-valued functions on (a, b) such that f(r) = g(r) for each rational number r in (a, b). Prove that f(x) = g(x) for all x in (a, b).

    2. Relevant equations

    3. The attempt at a solution

    Alright, well I'm thinking I should approach both of these problems in pretty much the same way. I know the following theorem, but I'm not sure where to go from here: "f is continuous at x_0 in dom(f) iff for each epsilon > 0 there exists delta > 0 such that x in dom(f) and |x - x_0| < delta imply |f(x) - f(x_0)| < epsilon.

    Any ideas? Thanks!
     
  2. jcsd
  3. Mar 26, 2008 #2

    Dick

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    For any delta>0 there is a rational number in the interval |x-x0|<delta. What does that tell you about f(x0)?
     
  4. Mar 26, 2008 #3

    HallsofIvy

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    Another way to do this is to use the fact that [itex]\lim_{x\rightarrow a} f(x)= \lim_{n\rightarrow \infty} f(x_n)[/itex] where [itex]{x_n}[/itex] is any sequence of numbers converging to a. In particular, there always exist a sequence of rational numbers converging to a.
     
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