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

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 continuous real-valued function

Proof

From Absolute Value of Function is Composite with Absolute Value Function:

$\size{f} = \size{\,\cdot\,} \circ f$

where:

$\size{\,\cdot\,}$ denotes the absolute value function $\size{\,\cdot\,} : \R \to \R$

$\size{\,\cdot\,} \circ f$ denotes the composite mapping of $\size{\,\cdot\,}$ with $f$

From Absolute Value Function is Continuous:

$\size{\,\cdot\,} : \R \to \R$ is continuous

From Composite of Continuous Mappings is Continuous:

$\size{\,\cdot\,} \circ f$ is continuous

The result follows.

$\blacksquare$