Homework 4 Solution

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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

0

(x irrational),

f (x) =

1

m

x =

.

n

n

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 .)

4


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