Combination Theorem for Bounded 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 $f :S \to \R$ be a bounded continuous real-valued function.

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 bounded continuous real-valued function

Proof

Follows from:

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

$\blacksquare$