Now that I have finished with the 7 chapters of rudin i plan to study Part 1 of royden's real analysis. Here is a problem regarding borel sets(adsbygoogle = window.adsbygoogle || []).push({});

Let f be a real valued function defined for all reals. Prove that the set of points at which f is continuous can be written as a countable intersection of open sets.

set y=f(x)

Consider the family of open sets, N_{1/n} (y) for every real y, call it V_n (y). Taking preimages of these sets will yield some open sets and some not. Let O_n be the union of the collection of preimages that are open, then O_n is open. Put

O=intersection of O_n.

If f is continuous at x the preimage of every V_n(f(x)) is open so it is contained in O_n which means that x is in O. Now if x is in O then x is in O_n for every n. Let e>0, pick a N so that 1/N<e/2. Then x is inside the preimage of some V_N (z). Since this the preimage of this V_N (z) is open, call it N(z), there exists a delta>0 so that if d(x,y)<delta then y is in N(z). Also

|f(x)-f(y)| <\= |f(x)-f(z)|+|f(y)-f(z)|<e

we have shown O is the set of points where f is continuous.

Can someone tell me if I'm missing something, i just feel like i am.

cheers!

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# The points at which a f is continuous is a G-delta

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