# Combination Theorem for Continuous Mappings/Normed Division Ring

## Theorem

Let $\struct{S, \tau_{_S}}$ be a topological space.

Let $\struct{R, +, *, \norm{\,\cdot\,}}$ be a normed division ring.

Let $\tau_{_R}$ be the topology induced by the norm $\norm{\,\cdot\,}$.

Let $\lambda \in R$.

Let $f,g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ be continuous mappings.

Let $U = S \setminus \set{x : \map g x = 0}$

Let $g^{-1} : U \to R$ denote the mapping defined by:

$\forall x \in U : \map {g^{-1}} x = \map g x^{-1}$

Let $\tau_{_U}$ be the subspace topology on $U$.

Then the following results hold:

### Sum Rule

$f + g: \struct {S, \tau_{_S} } \to \struct{R, \tau_{_R} }$ is continuous.

### Translation Rule

$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.

### Negation Rule

$- g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is continuous.

### Product Rule

$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.

### Multiple Rule

$\lambda * f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous
$f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.

### Inverse Rule

$g^{-1}: \struct {U, \tau_{_U} } \to \struct {R, \tau_{_R} }$ is continuous.