Combination Theorem for Continuous Real-Valued Functions/Negation Rule

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Theorem

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

Let $\R$ denote the real number line.

Let $f, g :S \to \R$ be continuous real-valued functions.

Let $-f : S \to \R$ denote the pointwise negation of $f$, that is, $-f$ denotes the mapping defined by:

$\forall s \in S : \map {\paren{-f} } s = - \map f s$

Then:

$-f$ is a continuous real-valued function

Proof

Follows from:

Real Numbers form Valued Field

By definition a valued field is a normed division ring

Negation Rule for Continuous Mappings into Normed Division Ring

$\blacksquare$