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 squash@math.ufl.edu 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 ( exer., or see notes- AdvCalc.pdf ). Were Cty( f ) = Q , it would be a dense G , 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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This note was uploaded on 01/26/2012 for the course MAC 3472 taught by Professor Jury during the Fall '07 term at University of Florida.

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