What are some standard methods of showing that a function element admits unrestricted continuation in a given domain ?
More specifically, a function element of $\cos^{-1}$ may be shown to exist in an $\varepsilon$ environment of the origin. (since $\cos '(\pi/2) \neq 0$ it's locally invertible, and the inverse is holomorphic.) How may this function element be shown to admit unrestricted continuation in $\mathbb{C} \setminus \{1,-1\}$ ?
