# Combination Theorem for Continuous Mappings

## Theorem

## Topological Semigroup

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

Let $\struct{G, *, \tau_{_G}}$ be a topological semigroup.

Let $\lambda \in G$.

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

Then the following results hold:

### Product Rule

- $f * g: \struct{S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.

### Multiple Rule

- $\lambda * f: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is a continuous mapping
- $f * \lambda: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is a continuous mapping.

## Topological Group

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

Let $\struct {G, *, \tau_{_G} }$ be a topological group.

Let $\lambda \in G$.

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

Then the following results hold:

### Product Rule

- $f * g : \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.

### Multiple Rule

- $\lambda * f: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping
- $f * \lambda: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.

### Inverse Rule

- $g^{-1}: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is a continuous mapping.

## Topological Ring

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

Let $\struct {R, +, *, \tau_{_R} }$ be a topological ring.

Let $\lambda \in R$.

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

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.

## Topological Division Ring

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

Let $\struct{R, +, *, \tau_{_R}}$ be a topological division ring.

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.

## Normed Division Rings

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.

## Metric Space

Let $M = \struct {A, d}$ be a metric space.

Let $\R$ denote the real numbers.

Let $f: M \to \R$ and $g: M \to \R$ be real-valued functions from $M$ to $\R$ which are continuous on $M$.

Let $\lambda, \mu \in \R$ be arbitrary real numbers.

Then the following results hold:

### Sum Rule

- $f + g$ is continuous on $M$.

### Difference Rule

- $f - g$ is continuous on $M$.

### Multiple Rule

- $\lambda f$ is continuous on $M$.

### Combined Sum Rule

- $\lambda f + \mu g$ is continuous on $M$.

### Product Rule

- $f g$ is continuous on $M$.

### Quotient Rule

- $\dfrac f g$ is continuous on $M \setminus \set {x \in A: \map g x = 0}$.

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

### Absolute Value Rule

- $\size f$ is continuous at $a$

where:

- $\map {\size f} x$ is defined as $\size {\map f x}$.

### Maximum Rule

- $\max \set {f, g}$ is continuous on $M$.

### Minimum Rule

- $\min \set {f, g}$ is continuous on $M$.

## Standard Number Fields

## 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 $\map g z \ne 0$.