To prove continuity of powers and roots of functions, some helpful rules are: if f is continuous (at a point or on an interval), and n is an integer, then f^n and f^(1/n) (the n-th root of f) are continuous (at that point or on that interval) wherever they are defined. So, if n is negative, f^n is continuous wherever f is nonzero and continuous. And if n is even, then f^(1/n) is continuous wherever f is positive and continuous. As a bonus, do you see how these rules are special cases of the fact that a composition of continuous functions is continuous?.