Let t denote the fractional part of t . Let f(x) = x^2 4 - x 2 be a function defined on the closed interval [0, 5] . Let S be the set of all points in [0, 5] at which f is not continuous. Then _ a S a^2 is equal to _______

Step-by-step solution

We know that t = t - [t] , where [t] is the greatest integer function. We can rewrite f(x) as: f(x) = ( x^2 4 - [ x^2 4 ] ) - ( x 2 - [ x 2 ] ) f(x) = ( x^2 4 - x 2 ) - ( [ x^2 4 ] - [ x 2 ] ) The polynomial part ( x^2 4 - x 2 ) is continuous everywhere. Therefore, the discontinuities of f(x) are exactly the discontinuities of g(x) = [ x^2 4 ] - [ x 2 ] . We find the points in [0, 5] where the inner functions are integers. x^2 4 is an integer for x^2 0, 4, 8, 12, 16, 20, 24, 25 . The points in [0, 5] are 0, 2, 8 ,