Problem 16 (Exercise 4.14). Let I = [0, 1] be the closed unit interval. Suppose f is a continuous mapping of I into I. Prove that f (x) = x for at least one x ∈ I.
Problem 17 (Exercise 4.18). Every rational x can be written in the form x = m/n, where n > 0, and m and n are integers without any common divisors. When x = 0, we take n = 1. Consider the function f defined on R1 by
f (x) =
Prove that f is continuous at every irrational point, and that f has a simple discontinuity at every rational point.
Problem 18 (Exercise 4.21). Suppose K and F are disjoint sets in a metric space X, K is compact,
is closed. Prove that there exists δ > 0 such that d(p, q) > δ if p ∈ K, q ∈ F . Show that the conclusion may fail for two disjoint closed sets if neither is compact.
Problem 19 (Exercise 5.1). Let f be defined for all real x, and suppose that
|f (x) − f (y)| ≤ (x − y)2
for all real x and y. Prove that f is constant.
Problem 20 (Exercise 5.3). Suppose g is a real function on R1, with bounded derivative (say
|g | ≤ M ). Fix 0, and define f (x) = x + (x). Prove that f is one-to-one if is small enough. (A set of admissible values of can be determined which depends only on M .)