Let f(x) = [x^2] and g(x) = 15 x , where [t] denotes the greatest integer t . Then, the number of points in the open interval (0, 2 ) where the composite function f(g(x)) is discontinuous is equal to ______.

Step-by-step solution

Given f(x) = [x^2] and g(x) = 15 x . The composite function is h(x) = f(g(x)) = [15 ^2 x] . Let u(x) = 15 ^2 x . The function u(x) oscillates between 0 and 15 . We analyze the number of points where u(x) takes integer values in the interval (0, 2 ) . In the interval (0, 2 ) , u(x) strictly decreases from 15 to 0 . It takes integer values 14, 13, , 1 , giving 14 points of discontinuity. At x = 2 , u( 2 ) = 0 . For x near 2 , u(x) > 0 , so [u(x)] = 0 . The value is 0 and the limit is 0 , so h(x) is continuous at x =