Combination Theorem for Continuous Real-Valued Functions

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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 $\lambda \in \R$.

Then the following results hold.

Sum Rule

$f + g$ is a coninuous real-valued function

Negation Rule

$-f$ is a continuous real-valued function

Difference Rule

$f - g$ is a coninuous real-valued function

Product Rule

$f g$ is a continuous real-valued function

Multiple Rule

$\lambda f$ is a continuous real-valued function

Absolute Value Rule

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

Maximum Rule

$f \vee g$ is a continuous real-valued function

Minimum Rule

$f \wedge g$ is a continuous real-valued function