Let [x] denotes the greatest integer less than or equal to x. If `f(x) = [x sin pix]`, then f(x) isA. continuous at x = 0B. continuous in `(-1, 0)`C. differentiable at x = 1D. differentiable in `(-1, 1)`

Let [x] denotes the greatest integer less than or equal to x. If `f(x)

1.	Let [x] denotes the greatest integer less than or equal to x. If `f(x) = [x sin pix]`, then f(x) isA. continuous at x = 0B. continuous in `(-1, 0)`C. differentiable at x = 1D. differentiable in `(-1, 1)`
Answer» Correct Answer - A::B::D We have, for ` -1 lt x lt 1` ` rArr" " [x tan pix] = 0` Also, x sin ` pi x` becomes negative and numerically less than 1 when x is slightly greater than 1 and so by definition of [x] . ` f(x) = [ x sin pi x ] =- 1, " when " 1 lt x lt 1 + h` Thus, f(x) is constant and equal to 0 in the closed interval `[-1, 1] ` and so f(x) is continuous and differentiable in the open interval ` (-1, 1)`. At x = 1, f(x) is discontinuous, since ` underset( h to 0) lim (1-h) = 0 ` and `underset( h to 0) lim (1+h) =- 1` `:.` f(x) is not differentiable at x = 1 . Hence, (a), (b) and (d) are correct answers.