1.) Let A and B be subsets of real numbers. Show that if there exist disjoint open sets U,V where A is a subset of U and B is a subset of V, then A and B are seperated.(adsbygoogle = window.adsbygoogle || []).push({});

In doing the proof of this, I have come to the conclusion that any 2 disjoint, open sets are seperated. However, I don't see this in my book or notes anywhere, so I don't think I can use it. I was wondering if someone could offer some guidance as to how to prove this otherwise?

2.) a.) Show that using definition 4.3.1 (I will state it below), that any function f with the integers as the domain is necessarily continuous everywhere.

b.) Show in general that if c is an isolated point of A which is a subset of the reals, then f: A -> R is continuous at c.

Definition 4.3.1 - A function f is continuous at a point c if, for all epsilon > 0, there exists a delta > 0 such that whenever |x-c|<delta, |f(x)-f(c)|<epsilon.

To be completely honest, I don't have even a remote clue on this one.

Thanks for any assistance, I appreciate it greatly.

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# Homework Help: Disjoint open sets homework

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