Continuidade e raizes nas definições de limite

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 $f^{1/n} = \sqrt[n]{f}$ 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 $\sqrt[n]{f}$ 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?