Let {r_n} be a listing of all the rational numbers and let {v_n} be a sequence of nonzero numbers that converges to 0. Define a function f by f(x)=0 if x is irrational and f(r_n)=v_n for all n. Show that f is continuous everywhere except for the set of rational numbers.

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.