Introduction to Mathematical Analysis I - Second Edition

160 Solutions and Hints for Selected Exercises SECTION 4.3 Exercise 4.3.2 . (a) We can prove that f is uniformly continuous on R by definition. Given any ε > 0, choose δ = ε ` + 1 1 α and get | f ( u ) − f ( v ) | ≤ ` | u − v | α < ` δ α = ` ε ` + 1 < ε whenever | u − v | < δ . Note that we use ` + 1 here instead of ` to avoid the case where ` = 0. (b) We will prove that f is a constant function by showing that it is differentiable on R and f 0 ( a ) = 0 for all a ∈ R . Fix any a ∈ R . Then, for x 6 = a , f ( x ) − f ( a ) x − a ≤ ` | x − a | α | x − a | = ` | x − a | α − 1 . Since α > 1, by the squeeze theorem, lim x → a f ( x ) − f ( a ) x − a = 0 . This implies that f is differentiable at a and f 0 ( a ) = 0. (c) We can verify that the function f ( x ) = | x | satisfies the requirement. From this problem, we see that it is only interesting to consider the class of functions that satisfy ( 4.8 ) when α ≤ 1. It is an exercise to show that the function f ( x ) = | x | 1 / 2 satisfies this condition with ` = 1 and α = 1 / 2. Exercise 4.3.3 . Define the function h ( x ) = g ( x ) − f ( x ) . Then h 0 ( x ) = g 0 ( x ) − f 0 ( x ) ≥ 0 for all x ∈ [ x 0 , ∞ ) . Thus, h is monotone increasing on this interval. It follows that h ( x ) ≥ h ( x 0 ) = g ( x 0 ) − f ( x 0 ) = 0 for all x ≥ x 0 . Therefore, g ( x ) ≥ f ( x ) for all x ≥ x 0 . Exercise 4.3.5 . Apply the mean value theorem twice. Exercise 4.3.6 . Use proof by contradiction. SECTION 4.4 Exercise 4.4.5 . Suppose that P ( x ) = a 0 + a 1 x + · · · + a n x n . Then apply L’Hospital’s rule repeatedly.