Question

Choose the most appropriate options. 
If \( f(x) = [x \sin n\pi x] \), then which of the following is incorrect?

• A. \( f(x) \text{ is continuous at } x = 0 \)
• B. \( f(x) \text{ is continuous in } (-1, 0) \)
• C. \( f(x) \text{ is differentiable at } x = 1 \)
• D. \( f(x) \text{ is differentiable in } (-1, 1) \)

Hint:

Always verify differentiability by checking the behavior around the point of interest for discontinuities.

Answer 1

The Correct Option is C

The function \( f(x) = [x \sin n\pi x] \) is continuous at \( x = 0 \) and differentiable in the intervals except at certain points.
However, it is not differentiable at \( x = 1 \), as the behavior at that point is not smooth due to the discontinuity caused by the sine function.