The number of points in the interval [2, 4] , at which the function f(x) = [x^2 - x - 1 2 ] , where [ ] denotes the greatest integer function, is discontinuous, is _______.

Step-by-step solution

Let g(x) = x^2 - x - 1 2 . Differentiating with respect to x , we get g'(x) = 2x - 1 . For x [2, 4] , g'(x) > 0 , which implies that g(x) is strictly increasing in the interval [2, 4] . The values of g(x) at the endpoints are: g(2) = 2^2 - 2 - 1 2 = 1.5 g(4) = 4^2 - 4 - 1 2 = 11.5 The function f(x) = [g(x)] is discontinuous at all points where g(x) is an integer, as g(x) is strictly monotonic and crosses these integer values. The integers between 1.5 and 11.5 are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 . Since g(x) is stric