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Let $f(x)=0$ if x is irrational and $f(p/q) = 1/q$ if $p$ and $q$ are positive integers with no common factors. Show that f is discontinuous at every rational and continuous at every irrational on $(0,\infty{})$

Let $x_0=p/q$ that an an element of interval I on which f is continouous. Why is it that $|f(x)-f(x_0)|=1/q$ means f is discontinuous at every rational number. Can't we just choose $\delta$ such that $1/q<\epsilon$

BCLC
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angelo086
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  • Use density of rational/irrational numbers. See http://math.stackexchange.com/questions/207118/prove-continuity-on-a-function-at-every-irrational-point-and-discontinuity-at-ev – charmd Oct 06 '16 at 19:48
  • http://math.stackexchange.com/questions/1895167/x-in-a-r-ar-versus-x-a-r-a http://math.stackexchange.com/questions/207118/prove-continuity-on-a-function-at-every-irrational-point-and-discontinuity-at-ev https://en.wikipedia.org/wiki/Thomae%27s_function http://math.stackexchange.com/questions/843704/showing-the-modified-dirichlet-function-is-discontinuous – BCLC Oct 11 '16 at 16:34

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