Combination Theorem for Continuous Mappings/Topological Division Ring/Product Rule

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

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

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

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

Then