Combination Theorem for Continuous Mappings/Topological Ring/Product Rule

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

Let $\struct{R, +, *, \tau_R}$ be a topological 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
 * $f * g : \struct{S, \tau} \to \struct{R, \tau_R}$ is continuous.

where $f * g : S \to R$ is the mapping defined by:
 * $\forall x \in S: \map {\paren{f * g}} x = \map f x * \map g x$