Combination Theorem for Bounded Continuous Real-Valued Functions/Absolute Value 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 bounded continuous real-valued functions.

Let $\size f : S \to \R$ denote the absolute value of $f$, that is, $\size f$ denotes the mapping defined by:

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

Then:

$\size f$ is a bounded continuous real-valued function

Proof

Follows from:

• Absolute Value Rule for Bounded Real-Valued Function
• Absolute Value Rule for Continuous Real-Valued Function

$\blacksquare$