MAN: Show that the function f of y equals 2x squared y cubed plus 3, and g of x y equal to parentheses 2x squared y cubed plus close parenthesis to the fourth are continuous everywhere. We're going to make an argument here for these functions. To begin with f of x equal to 2x squared is a continuous function, as is g of y equal to y cubed. And we know that the product of continuous functions is continuous. Therefore, f of xy is continuous and of course adding a constant does not affect that continuity.

Second, g of xy is continuous because we can think of this as a composition. Think of it as a composition of h of y equals y to the fourth, and we have f of xy being composed with h to be equal to g of x, y. And we know that the composition of two functions is continuous, therefore, f of xy and z of z are continuous everywhere.