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Proof about dense set and continuous function

  1. Feb 6, 2013 #1
    1. The problem statement, all variables and given/known data

    If f is continuous, and f(x) = 0 for all x in A, where A is a dense set. Then f(x) = 0 for all x.

    I am using the following definitions:
    A set of real numbers A is dense if every open interval contains a point of A.
    And the limit definition for a continuous function.

    2. Relevant equations



    3. The attempt at a solution

    Suppose there is an a such that f(a) > 0, then since f is continuous, there is an δ > 0, such that, for all x,
    if |x - a| < δ, then |f(x) - f(a)| < f(a). So f(x) > 0 for all x in (-δ + a, δ + a).
    But since (-δ + a, δ + a) is an open interval it contains a point of A call it z, but that is impossible, because then f(z) = 0 (By hypothesis), and f(z) > 0 because f(x) > 0 for all x in (-δ + a, δ + a). Assuming there is an a such that f(a) > 0 leads to a contradiction.

    I did basically the same to show that there are no x with f(x) < 0.
    Therefore f(x) = 0 for all x.

    Am I correct?

    Also, hello.
     
  2. jcsd
  3. Feb 6, 2013 #2

    micromass

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    That would be correct.

    As a matter of correctness though, you should really mention somewhere what the codomain and domain of f are. I assume here that you're working with [itex]f:\mathbb{R}\rightarrow \mathbb{R}[/itex].
     
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