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Thread: Continuous functions

  1. #1
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    Continuous functions

    Suppose that f : (a, b) R is continuous and that f(r) = 0 for ever rational number r є (a, b). Prove that f(x) = 0 for all x є (a, b).

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  2. #2
    Junior Member Infophile's Avatar
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    Hello,

    For all $\displaystyle x\in [a,b]$ reals, it exists a sequence of rationals $\displaystyle (r_n)$ such as $\displaystyle r_n\longrightarrow x$.

    Since $\displaystyle f$ is continuous then $\displaystyle f(r_n)=0\longrightarrow f(x)$

    And so...

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  3. #3
    MHF Contributor

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    Very nice- and not at all what I was thinking of. I was thinking of a proof by contradiction. Suppose there exist some a such that $\displaystyle f(a)= b\ne 0$. Let $\displaystyle \epsilon= |b|/2$. Then for all $\displaystyle \delta> 0$, there exist rational x in the interval $\displaystyle (a-\delta, a+ delta)$. For that x, $\displaystyle |x- a|< \delta$ but $\displaystyle |f(x)- f(a)|= |0- b|= |b|> |b|/2= \epsilon$, contradicting the fact that f is continuous.
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