Suppose that exist and is positive. Show that for any positive integer n we have
.
Those are meant to be n'th roots, right? ?
The slick way to do this is to say that is just the composition of the function f with the n'th root function . The function g(t) is continuous at all points t>0 (because in fact it is differentiable, with derivative ). And the composition of two continuous functions is continuous. So the function is continuous at (where is defined to be ).