Let [t] denote the greatest integer less than or equal to t . Let f(x) = [x^2] - [2x] be a function defined on the open interval (0, 2) . The number of points in the interval (0, 2) at which f is not continuous is _______

Step-by-step solution

To find the points of discontinuity of f(x) = [x^2] - [2x] in (0, 2) , we first identify the points where the inner functions x^2 and 2x become integers. For x (0, 2) , x^2 (0, 4) . The integer values of x^2 occur at x^2 = 1, 2, 3 , which gives x = 1, 2 , 3 . For x (0, 2) , 2x (0, 4) . The integer values of 2x occur at 2x = 1, 2, 3 , which gives x = 0.5, 1, 1.5 . The possible points of discontinuity are x = 0.5, 1, 2 , 1.5, 3 . Notice that x = 1 is a common point. We check the continuity of f(x) at x = 1 : Left-han