Combination Theorem for Continuous Mappings

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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 $\map g z \ne 0$.