27 4 Chapter 5 • Continuous Functions
EXERCISE SET 5.5- Prove Lemma 5.5.1.
- Prove that a function f is monotone (or strictly) increasing on an interval
I if and only if -f is monotone (or strictly) decreasing on I. - Suppose f and g are monotone increasing on an interval I.
(a) Prove that f + g is monotone increasing on I. [Strictly increasing if
f and g are strictly increasing.]
(b) Show by example that f g need not be monotone on I.
(c) Prove that if f , g are nonnegative on I , then Jg is monotone increas-
ing on I. [Strictly increasing if f , g are positive and strictly increasing
on J.]- Suppose f is continuous on an interval I and x 0 is an interior point of I
such that f(xo) = max{f(x) : x E I}. Prove that fir cannot be 1-1. [Of
course, the same conclusion holds if f(xo) = min{f(x) : x E J}.] - Find an example of a function f : [O, l] _, [O, l] that is 1-1 and onto but
not monotone on any (a, b) ~ I where a < b. - Complete Case 3 of the proof of Theorem 5.5.2.
- Prove Theorem 5.5.2 in the case where f is monotone decreasing on I.
- Prove Corollary 5.5.3 for the case in which f is strictly decreasing.
- Complete the proof of Claim #1 in the proof of Theorem 5.5.4 by con-
sidering the case in which f(b) < min{f(a), f(c)}. - Cantor Function: Find the numbers x 1 , x 2 , · · · , x 14 that are the end-
points of the open intervals removed from C 3 to create C 4. Calculate
'Pc(xi ) for i = 1, 2, · · · 14 and verify that 'Pc takes on the same value at
both endpoints of each of these removed intervals. - Prove that the Cantor function cp : [O, l] , [O, l] can b e extended to a
function cp : IR , IR that is continuous and monotone increasing, but not
strictly increasing on any nonempty open interval.
12. (Project) Positive Integral Power Functions: For a given n E N,
the function f(x) = xn is continuous on IR [see Theorem 5.1.7]. Prove
that(a) f is positive and strictly increasing on (0, +oo). [See Exercise 1.3.19.]