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 $\map g z \ne 0$.