Pointwise Addition on Continuous Real Functions forms Group

Theorem
Let $C$ be the set of all continuous real functions on the set of real numbers $\R$.

Let $f, g \in C$.

Let $f + g$ be the pointwise sum of $f$ and $g$:
 * $\forall x \in R: \left({f + g}\right) \left({x}\right) = f \left({x}\right) + g \left({x}\right)$

Then $\left({C, +}\right)$, the algebraic structure on $C$ induced by $+$, forms a group.

Proof
Taking the group axioms in turn:

G0: Closure
From the Combination Theorem for Functions, if $f$ and $g$ are continuous real functions then so is $f + g$.

Thus Closure is demonstrated.

G1: Associativity
Let $f, g, h \in C$.

From Real Addition is Associative, it follows directly that:
 * $\forall x \in \R: f \left({x}\right) + \left({g \left({x}\right) + h \left({x}\right)}\right) = \left({f \left({x}\right) + g \left({x}\right)}\right) + h \left({x}\right)$

so proving Associativity

G2: Identity
The constant function $f_0$ defined as:
 * $\forall x \in \R: f_0 \left({x}\right) = 0$

fulfils the role of the Identity:


 * $\forall x \in \R: f_0 \left({x}\right) + f \left({x}\right) = 0 + f \left({x}\right) = f \left({x}\right) = f \left({x}\right) + 0 = f \left({x}\right) = f_0 \left({x}\right)$

Note that $f_0 \in C$ as the Constant Function is Uniformly Continuous, and hence continuous.

G3: Inverses
From the Combination Theorem for Functions, if $f \left({x}\right)$ is continuous then so is $g \left({x}\right)$ where:
 * $\forall x \in \R: g \left({x}\right) = - f \left({x}\right)$.

Then we note that:
 * $\forall x \in \R: f \left({x}\right) + \left({- f \left({x}\right)}\right) = 0 = \left({- f \left({x}\right)}\right) + f \left({x}\right)$

So every element has an inverse.

All the group axioms are satisfied, hence the result.