Math 447
Extra Homework 3
Due May 2, 2006, before class
Consider (X, d) a metric space and Ea subset of X.
Definition 1 A set UโEis called open in Eif there exists an open set
U1โXsuch that U=U1TE.
Definition 2 A set FโEis called closed in Eif there exists a closed set
F1โXsuch that F=F1TE.
1. (20 points) Let (X, d),(Y, dโ) be two metric spaces and consider f:
XโYcontinuous and bijective.
(i) if Xis compact show that the inverse of fis continuous.
(ii) Let X=R, d(x, y) = ๎0x=y
1x6=yand Y=R, d โ(x, y) =
|xโy|.Show that (X, d) is a complete metric space and f:Xโ
Y, f (x) = xis continuous and bijective but its inverse is not
continuous.
2. (40 points) Let (X, d) be a metric space and EโX. Show that:
(a) If Eis connected then the empty set and Eare the only subsets
of Ethat are both open and closed in E;
(b) If Eis open and connected then Eis path connected. (Hint: Fix
a point xโEand show that the set of points in Ethat can be
connected to xby a path is both open and closed in E.)
3. (20 points) Find a counterexample to the following โTheoremโ:
Let A, B โRand f:AโBbe bijective and differentiable at x0โ
IntA. Then the inverse of fis differentiable at f(x0).
4. (20 points) Show that if fn:RโRis a sequence of differentiable
functions such that
