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no_fnc_cts_on_Q - There does not exist f R continuous...

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Unformatted text preview: There does not exist f : R continuous exactly on Q : Topology,BCT Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ ∼ squash/ 2 May, 2009 (at 15:45 ) Abstract: Gives various applications of BCT , the Baire Category theorem . Note. The ruler function R : R → [ , 1 ] , R ( x ) := if x irrational; 1 q if x has form p q in lowest terms is continuous precisely on the irrationals. The next thm shows that the opposite of this behavior is not possible. 1 : Theorem. There does not exist a function f : R → R with Cty( f ) = Q . ♦ Pf. The set Cty( f ) is always a G δ ( exer., or see notes- AdvCalc.pdf ). Were Cty( f ) = Q , it would be a dense G δ , hence residual. But Q , being countable, is meager. 2 : Theorem. Suppose we have sets A,B ⊂ R , each R-dense, and continuous functions f n : R → R such that f n A n →∞---→ A and f n B n →∞---→ 1 B , † : where each convergence is pointwise. Then thiswhere each convergence is pointwise....
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