Question

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

(a) continuous at x = 0

(b) continuous in (-1, 0)

(c) differentiable at x = 1

(d) differentiable in (-1, 1)

Answer 1

Correct option 

(a) continuous at x = 0

(b) continuous in (-1, 0)

(d) differentiable in (-1, 1)

Explanation :

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]

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).