150 Chapter^3For example, an arbitrary open interval of the real line JR is homeomorphic
to JR.A mapping f: X ---7 Y is a local homeomorphism if f is one-to-one and
bicontinuous when restricted to some neighborhood of each point of X.EXERC][SES 3.2
- Discuss the continuity and uniform continuity of the following functions
on the domains given.
(a) f(z) = z^3 , D = {z: /z/ < 2}
1
(b) f(z) = -. , D = {z: /z/ < 1}
z+i
z+l
(c) f(z) = -, D = {z: /z/ > 1}
z - Show that w = z/(l + /z/) defines a one-to-one continuous mapping of
the complex plane onto /w/ < 1, thus furnishing a finite model of the
complex plane.- Suppose that limz->oo f(z) = L and that f is defined for every positive
integer n. Prove that limn->oo f(n) =: L. Show by example that the
converse does not hold. - Suppose that f and g are uniformly continuous on a set S. Are f + g,
f g and f / g(g =/= 0) uniformly continuous on S?
5. Suppose that f is continuous in C, and let B = {z: J(z) = O} (the set
of zeros of !). Prove that B is closed.
6. Let X and Y be topological spaces, and suppose that (1) f: X ---7 ontoy
and (2) f is locally a homeomorphism. Prove that f is an open mapping.
- Let f(z) = (Rez)//z/ and g(z) = (zRez)//z/ for z =/= 0. Show that the
definition off cannot be extended so as to make ~he function continuous
at z == 0, but that is can be done for g.
8. Let (X, d) and (Y, d') be two metric spaces. If f: X ---7 Y is uni-
formly continuous on X and A, B are nonempty subsets of X such
that d(A, B) = 0, show that d'(f(A), f(B)) = 0. Conversely, if for
every pair of nonempty subsets A, B of X, d( A, B) = 0 implies thatd'(f(A), f(B)) = O, then f is uniformly continuous on X.
R. Cleveland, [Amer. Math. Monthly, 80 (1973), 64-66].- A compact and connected set is called a continuum. Prove that the
image of a locally connected continuum under a continuous mapping is
also a locally connected continuum.
3.13 Oriented Arcs and Curves
Definition 3.16 An oriented (closed) arc 'Y is defined to be a continuous
function from a closed real interval [a, ,8] into a topological space S.