Combination Theorem for Continuous Mappings/Normed Division Ring/Sum Rule

Theorem
Let $T = \struct{S, \tau}$ be a topological space.

Let $U \subseteq S$ be an open set in $T$.

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

Let $f: U \to R$ and $g: U \to R$ be continuous mappings.

Then:
 * $f + g : U \to R$ is continuous.

where $f + g : U \to R$ is the mapping defined by:
 * $\forall x \in U: \map {\paren{f + g}} x = \map f x + \map g x$

Proof
Let $\tau_U$ be the subspace topology on $U$.

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

Let $\tau_{R \times R}$ be the product topology on $R \times R$.

Let $f \times g : \struct{U, \tau_U} \to \struct{R \times R, \tau_{R \times R}}$ be the mapping defined by:
 * $\forall x \in U : \map {\paren{f \times g}} x = \tuple{\map f x, \map g x}$

Let $+ : \struct{R \times R, \tau_{R \times R}} \to \struct{R, \tau_R}$ be the mapping defined by:
 * $\forall x, y \in R : \map + {x,y} = x + y$

Lemma

 * $f + g = + \circ \paren{f \times g}$

where $+ \circ \paren{f \times g}$ is the composition of

Lemma

 * $f \times g$ is continuous.

From Addition on Normed Division Ring is Continuous, the mapping $+ : \struct{R \times R, \tau_{R \times R}} \to \struct{R, \tau_R}$ defined by:
 * $\forall x, y \in R : \map + {x,y} = x + y$

is continuous.

From Composite of Continuous Mappings is Continuous, the composition $+ \circ \paren{f \times g}$ is continuous.

Now:
 * $\forall x \in U : \map {+ \circ \paren{f \times g}} x = \map + {\map {\paren{f \times g}} x} = \map + {\tuple{\map f x, \map g x}} = \map f x + \map g x$