ASSIGNMENT 8
MAT 472 ยทFALL 2005
Also look at: Problems 1โ12 from Chapter IV of Rosenlicht.
Problem 1 (See Problem IV.9).Consider the function f:RโRdefined by
f(x) = ๎1/q if x=p/q, where pโZ,qโN, and pand qare relatively prime
0 if xis not rational.
Determine precisely the set of points of Rat which fis continuous, and justify your answer.
Problem 2 (See Problem IV.3).Let (E, dE) and (F , dF) be metric spaces, let f:EโF
be a function, let Aand Bbe closed subsets of Esuch that AโชB=E, and let fA:AโF
and fB:BโFbe the restrictions of fto Aand B, respectively. Prove that fis continuous
if and only if both fAand fBare. Also show by example that the conclusion can fail if A
and Bare not both closed.
Problem 3 (See Problem IV.4).Let Uand Vbe (possibly unbounded) open intervals in R,
and let f:UโVbe a function which is strictly increasing and onto V. Prove that fis
continuous.
Problem 4 (See Problem IV.6).Let Ebe a metric space, let SโE, and let f:EโRbe
defined by
f(p) = ๎1 if pโS
0 if not.
(fis called the characteristic function of S.) Prove that the set of points of Eat which fis
not continuous is precisely the boundary of S. (You may use any definitions and facts about
boundaries that appear in Rosenlicht, my notes, and your homework.)
Date: October 11, 2005 / Due Date: Tuesday, October 18, 2005.
S. Kaliszewski, Department of Mathematics and Statistics, Arizona State University.
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