Question

Let [x] denote the greatest integer less than or equal to x. If f(x) = [x sinπ x], then f(x) is

• A. (A) continuous at x = 0
• B. (B) continuous in (-1, 0)
• C. (C) differentiable at x = 1
• D. (D) differentiable in (-1, 1)

Answer 1

Answer: (A), (D)

(A) continuous at x = 0

Answer 2

Explanation:
We have, for (-1∴[xsin⁡πx]=0Also, xsin⁡πx becomes negative and numerically less than 1 when x is slightly greater than 1 , and so by definition of [x] f(x)=[xsin⁡πx]=−1, when (1Thus, f(x) is constant and equal to 0 in the closed interval [-1,1] and s0f(x) is continuous and differentiable in the open interval (-1,1) At x=1,f(x) is discontinuous. since limi→0(1−h)=0 and limt→0(1+h)=−1∴f(x) is not differentiable at x=1 Hence, options (1),(2) and (4) are correct answers.