Let f(x) = [x] sin(πx). Is it differentiable at x = 1?

Let f(x) = [x] sin(πx). Is it differentiable at x = 1?

• A. Yes, f'(1) = 0
• B. Yes, f'(1) = π
• C. Yes, f'(1) = −π
• D. No, it is not differentiable

Answer: D) No, it is not differentiable

Explanation: LHD limit relies on [1−h] = 0, giving 0. RHD limit relies on [1+h] = 1, giving lim(h→0) sin(π+πh)/h = lim(h→0) −sin(πh)/h = −π. Since 0 ≠ −π, it is not differentiable.