# Combination Theorem for Continuous Functions/Quotient Rule

## Theorem

### Real Functions

$\dfrac f g$ 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

$\dfrac f g$ is continuous on $S \setminus \set {z \in S: \map g z = 0}$

that is, on all the points $z$ of $S$ where $\map g z \ne 0$.