Introduction to Mathematical Analysis I - Second Edition

157 From the solution, it is important to see that the conclusion remains valid if we replace the function f by g ( x ) =   x n sin 1 x , if x 6 = 0; 0 , if x = 0 , where n ≥ 2, n ∈ N . Note that the function h ( x ) = cx does not play any role in the differentiability of f . We can generalize this problem as follows. Let ϕ be a bounded function on R , i.e., there is M > 0 such that | ϕ ( x ) | ≤ M for all x ∈ R . Define the function f ( x ) = ( x n ϕ ( 1 / x ) , if x 6 = 0; 0 , if x = 0 , where n ≥ 2, n ∈ N . Then f is differentiable at a = 0. Similar problems: 1. Show that the functions below are differentiable on R : f ( x ) = ( x 3 / 2 cos ( 1 / x ) , if x ≥ 0; 0 , if x < 0 and f ( x ) = ( x 2 e − 1 / x 2 , if x 6 = 0; 0 , if x = 0 . 2. Suppose that ϕ is bounded and differentiable on R . Define the function f ( x ) = ( x n ϕ ( 1 / x ) , if x 6 = 0; 0 , if x = 0 . Show that if n ≥ 2, the function is differentiable on R and find its derivative. Show that if n = 1 and lim x → ∞ ϕ ( x ) does not exists, then f is not differentiable at 0. (b) Hint: Observe that f 0 1 2 n π = − 1 + c < 0 and f 0 1 ( 2 n + 1 ) π = 1 + c > 0 . SECTION 4.2 Exercise 4.2.1 . Define the function h ( x ) = f ( x ) − g ( x ) .