sequence of continuous functions

{f_n(x)} is a sequence of continuous functions on R^1 and it converges
pointwise everywhere to f(x).

Prove that for any G in R^1, f^-1(x) is the union of a countable
collection of closed sets. Furthermore, let C_f be the set of continous
points for f(x), show that C_f is dense in R^1.

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