For f(x) = x sin(1/x) (x ≠ 0) and f(0) = 0, which of the following is true at x = 0?

For f(x) = x sin(1/x) (x ≠ 0) and f(0) = 0, which of the following is true at x = 0?

• A. Continuous and Differentiable
• B. Continuous but Not Differentiable
• C. Not Continuous but Differentiable
• D. Neither Continuous nor Differentiable

Answer: B) Continuous but Not Differentiable

Explanation: It is continuous because lim(x→0) x sin(1/x) = 0 = f(0). However, f'(0) = lim(h→0) (h sin(1/h))/h = lim(h→0) sin(1/h), which oscillates and does not exist.