Combination Theorem for Continuous Real-Valued Functions/Multiple Rule
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Theorem
Let $\struct{S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $f :S \to \R$ be contiuous real-valued function.
Let $\lambda \in \R$.
Let $\lambda f : S \to \R$ be the pointwise scalar multiplication of $f$ by $\lambda$, that is, $\lambda f$ is the mappping defined by:
- $\forall s \in S : \map {\paren{\lambda f} } s = \lambda \map f s$
Then:
- $\lambda f$ is a continuous real-valued function
Proof
Follows from:
- Real Numbers form Valued Field
- By definition a valued field is a normed division ring
- Multiple Rule for Continuous Mappings into Normed Division Ring
$\blacksquare$