On the possibility of division and involution to a fractional power in the algebra of rational functions

Suppose that a function $f(z)$ satisfies a Lipschitz condition with an arbitrary positive element on a compact set $X$ in $\mathbf C$ and can be uniformly approximated on $X$ by rational functions. If $q>1$ and some branch of $(f(z))^q$ is continuous on $X$, then this branch can also be approximated on $X$ by rational functions. Also, an example is given of a compact set $X$ and two functions $f(z)$ and $g(z)$ uniformly approximable on $X$ by rational functions and with ratio $g(z)/f(z)$ naturally (uniquely) defined and continuous on $X$ but not approximable by rational functions.
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