Combination Theorem for Continuous Mappings/Normed Division Ring/Inverse Rule

Theorem
Let $T = \struct{S, \tau}$ be a topological space.

Let $U \subseteq S$ be an open set in $T$.

Let $\struct{R, +, \circ, \norm{\,\cdot\,}}$ be a normed division ring.

Let $g: U \to R$ be a continuous mapping.

Let $U’ = U \setminus \set{x : \map g x = 0}$

Then:
 * $g^{-1} : U’ \to R$ is continuous.

where $g^{-1} : U’ \to R$ is the mapping defined by:
 * $\forall x \in U’ : \map {g^{-1}} x = \map g x^{-1}$