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**Real Analysis--Prove Continuous at each irrational and discontinuous at each rational**

The question is, Let {q1, q2...qn} be an enumeration of the rational numbers. Consider the function f(x)=Summation(1/n^2). Prove that f is continuous at each rational and discontinuous at each irrational.

Normally I would try to use Rudin's Th 7.11 (Principles of Mathematics) but I'm not sure if this still works with summation involved...

A very rough proof:

First of all, note that summation(1/n^2) is uniformly continuous by Th 7.10. Let x be an irrational, and x is continuous at all x by Th 7.11 so the sum is also continuous at x.

Is this correct?