Let [x] denote the greatest integer less than or equal to x . Let m and n be the number of points in the interval (-1, 3) where the function f(x) = ( x 2 ) [x] is discontinuous and non-differentiable, respectively. Then the value of m+n is :

Step-by-step solution

The given function is f(x) = ( x 2 ) [x] . The potential points of discontinuity and non-differentiability in the interval (-1, 3) are the integers x = 0, 1, 2 . Checking continuity at x = 0 : _ x 0^- f(x) = (0) (-1) = -1 _ x 0^+ f(x) = (0) 0 = 0 Since the left-hand limit and right-hand limit are not equal, f(x) is discontinuous at x = 0 . Checking continuity at x = 1 : _ x 1^- f(x) = ( 2 ) 0 = 0 _ x 1^+ f(x) = ( 2 ) 1 = 0 Also f(1) = 0 . Thus, f(x) is continuous at x = 1 . Checking continuity at x = 2 : _ x 2^- f(