f(x) = { x² sin(1/x), x ≠ 0; 0, x = 0 }. Which statement is correct?

f(x) = { x² sin(1/x), x ≠ 0; 0, x = 0 }. Which statement is correct?

• A. f is differentiable at 0 and f′(0)=0
• B. f is differentiable at 0 and f′(0)=1
• C. f is continuous but not differentiable at 0
• D. f is discontinuous at 0

Answer: A) f is differentiable at 0 and f′(0)=0

Explanation: lim(x→0) x² sin(1/x) = 0 = f(0), so continuous. f′(0) = lim(h→0) h² sin(1/h)/h = lim h sin(1/h) = 0. So differentiable at 0 with derivative 0.