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

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

Let $U \subseteq S$ be an open set in $T$.

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

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

Let $\lambda \in R$.

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

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