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I can find that the function f \begin{align} f(x)= & 1/p & (x=p/q, p/q ~ simple) \\ { } & 0 & (x ~ is ~ irrational) \end{align} is continuous only on x for x irrational and discontinuous on x for x rational. However, I can't find the measurable function that is discontinuous on irrational x and continuous on rational x. So I raise question whether the measurable function f that is continuous on $B$ exists for any Borel set $B$. If it is not true, what is the condition for $B$ that allows a measurable function f continuous only on $B$ and discontinues on $B^c$?

Lee Yohan
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    The rationals are not a $G_\delta$, so cannot be the set of points of continuity. I think this has appeared a number of times on MSE. – André Nicolas Sep 21 '13 at 05:21

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