Combination Theorem for Continuous Mappings/Normed Division Ring/Translation Rule

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

Let $\struct {R, +, *, \norm {\,\cdot\,} }$ be a normed division ring.

Let $\tau_{_R}$ be the topology induced by the norm $\norm {\,\cdot\,}$.

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:
 * $\forall x \in S: \map {\paren {\lambda + f} } x = \lambda + \map f x$

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

Proof
From Corollary to Normed Division Ring Operations are Continuous:
 * $\struct{R, +, *, \tau_{_R} }$ is a topological division ring.

From Translation Rule for Continuous Mappings to Topological Division Ring:
 * $\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.

Also see

 * Translation Rule for Continuous Mappings to Topological Division Ring