Let f (x) = x 2 sin (1/x) for x ≠ 0 and f (0) = 0. (a) Use the chain rule and the product rule to show that f is differentiable at each c ≠ 0 and find f ′(c). (You may assume that the derivative of...

Let f (x) = x²
sin (1/x) for x ≠ 0 and f (0) = 0.

(a) Use the chain rule and the product rule to show that f is differentiable at each c ≠ 0 and find f ′(c). (You may assume that the derivative of sin x is cos x for all x ∈
.)

(b) Use Definition 1.1 to show that f is differentiable at x = 0 and find f ′(0).

(c) Show that f ′ is not continuous at x = 0.

(d) Let g(x) = x²
if x ≤ 0 and g(x) = x²
sin (1/x) if x > 0. Determine whether or not g is differentiable at x = 0. If it is, find g′(0)