Introduction to Mathematical Analysis I - Second Edition

83 (a) Prove that the equation f ( x ) = x has a solution on [ a , b ] . (b) Suppose further that | f ( x ) − f ( y ) | < | x − y | for all x , y ∈ [ a , b ] , x 6 = y . Prove that the equation f ( x ) = x has a unique solution on [ a , b ] . 3.4.6 B Let f be a continuous function on [ a , b ] and x 1 , x 2 , . . . , x n ∈ [ a , b ] . Prove that there exists c ∈ [ a , b ] with f ( c ) = f ( x 1 )+ f ( x 2 )+ · · · f ( x n ) n . 3.4.7 B Suppose f is a continuous function on R such that | f ( x ) | < | x | for all x 6 = 0 . (a) Prove that f ( 0 ) = 0. (b) Given two positive numbers a and b with a < b , prove that there exists ` ∈ [ 0 , 1 ) such that | f ( x ) | ≤ ` | x | for all x ∈ [ a , b ] . 3.4.8 I Let f , g : [ 0 , 1 ] → [ 0 , 1 ] be continuous functions such that f ( g ( x )) = g ( f ( x )) for all x ∈ [ 0 , 1 ] . Suppose further that f is monotone. Prove that there exists x 0 ∈ [ 0 , 1 ] such that f ( x 0 ) = g ( x 0 ) = x 0 . 3.5 UNIFORM CONTINUITY We discuss here a stronger notion of continuity. Definition 3.5.1 Let D be a nonempty subset of R . A function f : D → R is called uniformly continuous on D if for any ε > 0, there exists δ > 0 such that if u , v ∈ D and | u − v | < δ , then | f ( u ) − f ( v ) | < ε . Example 3.5.1 Any constant function f : D → R , is uniformly continuous on its domain. Indeed, given ε > 0, | f ( u ) − f ( v ) | = 0 < ε for all u , v ∈ D regardless of the choice of δ . The following result is straightforward from the definition. Theorem 3.5.1 If f : D → R is uniformly continuous on D , then f is continuous at every point x 0 ∈ D . Example 3.5.2 Let f : R → R be given by f ( x ) = 7 x − 2. We will show that f is uniformly continuous on R . Let ε > 0 and choose δ = ε / 7. Then, if u , v ∈ R and | u − v | < δ , we have | f ( u ) − f ( v ) | = | 7 u − 2 − ( 7 v − 2 ) | = | 7 ( u − v ) | = 7 | u − v | < 7 δ = ε .