question_answer 1)

Show that g(x) = x – [x] is discontinuous at all integral points. Where [x] is greatest integer function.  

Answer:

Let g(x) = f(x) ? h(x)       f(x) = x, a polynomial function, is continuous function and h(x) ? h(x)       f(x) = x, a polynomial function, is continuous function and h(x) = [x]       Let a is any integer       At x = a                                     (by definition)                                     (by definition)       Since       Therefore h(x) is discontinuous at x = a which is an integral point.                        Now g(x) = f(x) ? h(x) will be continuous, if both f and h are continuous, but here h is not continuous at integral points. Hence g is discontinuous at integral points.