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