256 Chapter 5 11 Continuous Functions
- Prove that if f : I--+ JR. is continuous on an interval I , and f (I) contains
only rational numbers, then f is constant on I. Can the same statement
be proved if the rational numbers are replaced by irrational numbers?
What if the rational numbers are replaced by any set whose complement
is dense in JR.? - Prove Corollary 5.3.14.
- Suppose f: lR.-+lR. is continuous and lim f(x) = lim f(x) = 0. Prove
x--+-oo x --++oo
that f is bounded on JR., and attains either a maximum value or a mini-
mum value on JR., but not necessarily both.
23. Prove that if f : A-+JR. is continuous and 1-1 on a compact set A, then
1-^1 : f (A)-+A is continuous. [Hint: Use sequences and Exercise 2.6.21.]- Suppose f : [O, 27r] --+ JR. is continuous, and f(O) = f(27r). Prove that
there exists at least one point c E [O, 7r] such that f(c) = f(c+7r). [Hint:
Consider g(x) = f(x) - f(x + 7r).] Explain how from this result you can
conclude that on any great circle around the earth, there are at least
two diametrically opposite points at which the temperature is exactly the
same. - Suppose f: [a, b] --+JR. is continuous and f(a) = f(b). Prove that
(a) 3 x, y E [a, b] 3 y - x = ~(b - a) and f(x) = f(y), and hence,
(b) Ve> 0, 3x, y E [a,b] 3 ly-xl <€and f(x) = f(y).- Suppose f:[a, b] --+JR. is continuous and a::::; s < t::::; b. Prove that
Vm, n > 0, 3 c E (s, t) 3 f(c) = mf(s) + nf(t).
m+n- Characterization of Compact Sets: Prove that a nonempty set A
is compact if and only if every continuous f : A --+ JR. has the extreme
value property on A. [Hint: Somewhere in the proof a function of the form
1/lx -xol may be helpful.]
28. Characterization of Intervals: We say that a function f : A-+JR. has
the intermediate value property if for all a < b in A, Vy between f (a)
and f(b), 3c EA 3 f(c) = y. Prove that a set A is an interval if and only
if every continuous f : A-+JR. has the intermediate value property.5.4 Uniform Continuity
In a one-semester course, this section may be postponed until
Section 7.2, when Theorem 5 .4.7 is needed to prove Theorem
7.2.17.