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Math Help - Continuity question on rationals/irrationals

  1. #1
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    Continuity question on rationals/irrationals

    f:[0,2pi] -> R with f(x)= 0 for x in Q and f(x)=sin(x) for x in R\Q.

    At which points c in [0,2pi] is f continuos?

    I'm thinking it's discontinuous everywhere except 0,pi and 2pi.

    I've come up with this to show discontinuity at the rational points in [0,2pi]

    Consider a sequence x_n \to q \in \mathbb{Q} with  x_n \notin  \mathbb{Q} for every natural n. Then f(x_n) = sin(x_n) does not converge so discontinuous for every  c \in [0,2pi] s.t c is rational.

    But I'm not sure about how to prove the discontinuity/continuity at other points?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint: Every interval of the real line contains infinite rationals and infinite irrationals.
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