Combination Theorem for Continuous Mappings/Topological Ring/Multiple Rule

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

Let $\struct{R, +, *, \tau_{_R}}$ be a topological ring.

Let $\lambda \in R$.

Let $f: \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ be a continuous mapping.

Let $\lambda * f : S \to R$ be the mapping defined by:
 * $\forall x \in S: \map {\paren{\lambda * f}} x = \lambda * \map f x$

Let $f * \lambda : S \to R$ be the mapping defined by:
 * $\forall x \in S: \map {\paren{f * \lambda}} x = \map f x * \lambda$

Then
 * $\lambda * f : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is continuous.
 * $f * \lambda: \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is continuous.

Proof
By definition of a topological ring, $\struct{R, *, \tau_{_R}}$ is a topological semigroup.

From Multiple Rule for Continuous Mappings to Topological Semigroup, $\lambda * f, f * \lambda : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ are continuous mappings.

Also see

 * Multiple Rule for Continuous Mappings to Topological Semigroup