Combination Theorem for Continuous Mappings/Topological Division Ring

Theorem
Let $\struct{S, \tau_S}$ be a topological space.

Let $\struct{R, +, *, \tau_R}$ be a topological division ring.

Let $\lambda \in R$.

Let $f,g : \struct{S, \tau_S} \to \struct{R, \tau_R}$ be continuous mappings.

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

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

Let $\tau_U$ be the subspace topology on $U$.

Then the following results hold: