Combination Theorem for Bounded 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 bounded continuous real-valued functions.
Let $\lambda \in \R$.
Then the following results hold.
Sum Rule
- $f + g$ is a bounded coninuous real-valued function
Negation Rule
- $-f$ is a bounded continuous real-valued function
Difference Rule
- $f - g$ is a bounded coninuous real-valued function
Product Rule
- $f g$ is a bounded continuous real-valued function
Multiple Rule
- $\lambda f$ is a bounded continuous real-valued function
Absolute Value Rule
- $\size f$ is a bounded continuous real-valued function
Maximum Rule
- $f \vee g$ is a bounded continuous real-valued function
Minimum Rule
- $f \wedge g$ is a bounded continuous real-valued function