Let f:[-1, 2] R be a function defined by f(x) = x^2[x] - a x [x^2] , where a is a real constant and [t] denotes the greatest integer less than or equal to t . If f(x) is continuous at x = 1 , then the number of points in the interval [-1, 2] where f(x) is discontinuous is :

Step-by-step solution

Given f(x) = x^2[x] - a x [x^2] . Since f(x) is continuous at x=1 , we have _ x 1^- f(x) = _ x 1^+ f(x) = f(1) . Left-hand limit (LHL) at x=1 : As x 1^- , [x] = 0 and [x^2] = 0 . _ x 1^- f(x) = (1)^2(0) - a(1)(0) = 0 . Right-hand limit (RHL) at x=1 : As x 1^+ , [x] = 1 and [x^2] = 1 . _ x 1^+ f(x) = (1)^2(1) - a(1)(1) = 1 - a . Equating LHL and RHL, we get 0 = 1 - a a = 1 . Now, the function becomes f(x) = x^2[x] - x[x^2] . The doubtful points for discontinuity in [-1, 2] are the integers and the points where x^2 i