Combination Theorem for Bounded 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 bounded 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 bounded continuous real-valued function

Proof

Follows from:

• Multiple Rule for Bounded Real-Valued Function
• Multiple Rule for Continuous Real-Valued Function

$\blacksquare$