Introduction to Mathematical Analysis I - Second Edition

76 3.3 CONTINUITY Exercises 3.3.1 Prove, using definition 3.3.1 , that each of the following functions is continuous on the given domain: (a) f ( x ) = ax + b , a , b ∈ R , on R . (b) f ( x ) = x 2 − 3 on R . (c) f ( x ) = √ x on [ 0 , ∞ ) . (d) f ( x ) = 1 x on R \ { 0 } . 3.3.2 Determine the values of x at which each function is continuous. The domain of all the functions is R . (a) f ( x ) =   sin x x , if x 6 = 0; 1 , if x = 0 . (b) f ( x ) =   sin x | x | , if x 6 = 0; 1 , if x = 0 . (c) f ( x ) =   x sin 1 x , if x 6 = 0; 0 , if x = 0 . (d) f ( x ) =   cos π x 2 , if | x | ≤ 1; | x − 1 | , if | x | > 1 . (e) f ( x ) = lim n → ∞ sin π 2 ( 1 + x 2 n ) , x ∈ R . 3.3.3 Let f : R → R be the function given by f ( x ) = ( x 2 + a , if x > 2; ax − 1 , if x ≤ 2 . Find the value of a such that f is continuous. 3.3.4 Let f : D → R and let x 0 ∈ D . Prove that if f is continuous at x 0 , then | f | is continuous at this point. Is the converse true in general? 3.3.5 Prove Theorem 3.3.3 . ( Hint: treat separately the cases when x 0 is a limit point of D and when it is not.) 3.3.6 Prove parts (b) and (c) of Theorem 3.3.4 . 3.3.7 Prove Theorem 3.3.5 . 3.3.8 I Explore the continuity of the function f in each case below.