Combination Theorem for Continuous Functions
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Theorem
Real Functions
Let $\R$ denote the real numbers.
Let $f$ and $g$ be real functions which are continuous on an open subset $S \subseteq \R$.
Let $\lambda, \mu \in \R$ be arbitrary real numbers.
Then the following results hold:
Sum Rule
- $f + g$ is continuous on $S$.
Difference Rule
- $f - g$ is continuous on $S$.
Multiple Rule
- $\lambda f$ is continuous on $S$.
Combined Sum Rule
- $\lambda f + \mu g$ is continuous on $S$.
Product Rule
- $f g$ is continuous on $S$
Quotient Rule
- $\dfrac f g$ is continuous on $S \setminus \set {x \in S: \map g x = 0}$
that is, on all the points $x$ of $S$ where $\map g x \ne 0$.
Complex Functions
Let $\C$ denote the complex numbers.
Let $f$ and $g$ be complex functions which are continuous on an open subset $S \subseteq \C$.
Let $\lambda, \mu \in \C$ be arbitrary complex numbers.
Then the following results hold:
Sum Rule
- $f + g$ is continuous on $S$.
Multiple Rule
- $\lambda f$ is continuous on $S$.
Combined Sum Rule
- $\lambda f + \mu g$ is continuous on $S$.
Product Rule
- $f g$ is continuous on $S$
Quotient Rule
- $\dfrac f g$ is continuous on $S \setminus \set {z \in S: \map g z = 0}$
that is, on all the points $z$ of $S$ where $\map g z \ne 0$.