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

Theorem
Let $T = \struct{S, \tau}$ be a topological space.

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

Let $f: T \to R$ be a continuous mapping.

Let $\lambda \in R$.

Then:
 * $\lambda * f : T \to R$ is continuous.

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