Combination Theorem for Continuous Mappings/Normed Division Ring

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

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

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

Let $\lambda, \mu \in R$ be arbitrary elements in $R$.

Let $f: U \to R$ and $g: U \to R$ be continuous mappings.

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

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

Then the following results hold: