a) Show that the function h(x) defined by h(x) =x^4(sin(1/x)) for x not equal to 0; h(x)= 0 for x= 0 is continuous on (-infinity, infinity).

b) Consider the function g(x) = tan^-1(x)+x^3+ 1. Show that there exists c [1, 1] such that g(c) = 0.