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Math Help - Continuity

  1. #1
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    Continuity

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

    R - real numbers

    I need a proof of why this is true or if false a counterexample with proof.
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  2. #2
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    Quote Originally Posted by hayter221 View Post
    Let f, g be continous from R to R and suppose f(r)=g(r) for all rational numbers r. Is it true that f(x) = g(x) for all x in R.

    R - real numbers

    I need a proof of why this is true or if false a counterexample with proof.
    If x\in \mathbb{R} then there is a sequence \{ r_n \} with r_n \in \mathbb{Q} such that \lim r_n = x. Then by continuity f(x) = f(\lim r_n) = \lim f(r_n) and g(x) = g(\lim r_n) = \lim g(r_n). But f(r_n) = g(r_n). Thus, \lim f(r_n) = \lim g(r_n)\implies f(x) = g(x).
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