Introduction to Mathematical Analysis I - Second Edition

155 Therefore, {| f ( x n ) − f ( y n ) |} does not converge to 0 and, hence, f is not uniformly continuous on R . In this solution, we can use x n = r n + 1 n and y n = √ n for n ∈ N instead. (b) Use x n = 1 π / 2 + 2 n π and y n = 1 2 n π , n ∈ N . (c) Use x n = 1 / n and y n = 1 / ( 2 n ) . It is natural to ask whether the function f ( x ) = x 3 is uniformly continuous on R . Following the solution for part (a), we can use x n = 3 r n + 1 n and y n = 3 √ n for n ∈ N to prove that f is not uniformly continuous on R . By a similar method, we can show that the function f ( x ) = x n , n ∈ N , n ≥ 2, is not uniformly continuous on R . A more challenging question is to determine whether a polynomial of degree greater than or equal to two is uniformly continuous on R . Exercise 3.5.7 . Hint: For part (a) use Theorem 3.5.5 . For part (b) prove that the function can be extended to a continuous function on [ a , b ] and then use Theorem 3.5.5 . Exercise 3.5.8 . (a) Applying the definition of limit, we find b > a such that c − 1 < f ( x ) < c + 1 whenever x > b . Since f is continuous on [ a , b ] , it is bounded on this interval. Therefore, f is bounded on [ a , ∞ ) . (b) Fix any ε > 0, by the definition of limit, we find b > a such that | f ( x ) − c | < ε 2 whenever x > b . Since f is continuous on [ a , b + 1 ] , it is uniformly continuous on this interval. Thus, there exists 0 < δ < 1 such that | f ( u ) − f ( v ) | < ε 2 whenever | u − v | < δ , u , v ∈ [ a , c + 1 ] . Then we can show that | f ( u ) − f ( v ) | < ε whenever | u − v | < δ , u , v ∈ [ a , ∞ ) . (c) Since lim x → ∞ f ( x ) = c > f ( a ) , there exists b > a such that f ( x ) > f ( a ) whenever x > b . Thus, inf { f ( x ) : x ∈ [ a , ∞ ) } = inf { f ( x ) : x ∈ [ a , b ] } . The conclusion follows from the Extreme Value Theorem for the function f on [ a , b ] . SECTION 3.6 Exercise 3.7.4 . Since lim x → ∞ f ( x ) = lim x →− ∞ f ( x ) = ∞ , there exists a > 0 such that f ( x ) ≥ f ( 0 ) whenever | x | > a .