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Math Help - Continuity Question

  1. #1
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    Continuity Question

    Let f, g be continuous from \mathbb{R} to \mathbb{R} and suppose that f(r)=g(r) for all rational numbers r. Is it true that f(x)=g(x) for all x \in \mathbb{R}.

    I think yes, but am having trouble formulating a proof. Here is what I have so far.
    Assume for contradiction that it is not true. Then \exists m \in \mathbb{R}/ \mathbb{Q} s.t. f(m) \neq g(m), but from here I can't see where to get the contradiction. Any help?
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  2. #2
    Super Member girdav's Avatar
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    Re: Continuity Question

    Apply the definition of continuity with \varepsilon =\frac{|f(m)-g(m)|}2, and use the fact that there is a rational number in each open non empty interval.
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  3. #3
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    Re: Continuity Question

    Quote Originally Posted by worc3247 View Post
    Let f, g be continuous from \mathbb{R} to \mathbb{R} and suppose that f(r)=g(r) for all rational numbers r. Is it true that f(x)=g(x) for all x \in \mathbb{R}.
    Here is a second way.
    Every real number is the limit of a sequence of rational numbers.
    The functions f~\&~g agree on such a sequence.
    From continuity what can you conclude?
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