Let f(x) = { x sin(1/x) if x ≠ 0; 0 if x = 0 }. Which of the following statements is true?
Correct: A
First, check continuity at 0: |x sin(1/x)| ≤ |x| for all x ≠ 0. As x → 0, |x| → 0, so by the Squeeze Theorem, lim_(x→0) x sin(1/x) = 0 = f(0). Therefore f is continuous at 0. Now check differentiability: f'(0) = lim_(h→0) [f(h) - f(0)] / h = lim_(h→0) (h sin(1/h)) / h = lim_(h→0) sin(1/h). This limit does not exist because sin(1/h) oscillates between -1 and 1 as h → 0. Therefore f is not differentiable at 0. The correct answer is A.