# 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 $\map g z \ne 0$.