Group Homomorphism/Examples/Pointwise Addition on Continuous Real Functions on Closed Unit Interval/Example 10

Example of Group Homomorphism
Let $J \subseteq \R$ denote the closed unit interval $\closedint 0 1$.

Let $\map {\mathscr C} J$ denote the set of all continuous real functions from $J$ to $\R$.

Let $G = \struct {\map {\mathscr C} J, +}$ be the group formed on $\map {\mathscr C} J$ by pointwise addition.

Let $\struct {\R, +}$ denote the additive group of real numbers.

From Pointwise Addition on Continuous Real Functions on Closed Unit Interval forms Group we have that $G$ is indeed a group.

Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be the mapping defined as:
 * $\forall f \in \map {\mathscr C} J: \map \phi f = -\map f 0 + \ds \int_{-2}^0 \map f {e^x} \rd x$

Then $\phi$ is a homomorphism.

Proof
Let $f, g \in \map {\mathscr C} J$ be arbitrary.

We note that:
 * when $x = 0$ we have that $e^x = 1$
 * when $x = -2$ we have that $0 < e^x < 1$

Therefore the limits of integration of the given definition of $\phi$ are within $\closedint 0 1$.

As both $f$ and $g$ are continuous real functions on $J$, they are integrable on $\closedint {e^{-2} } 1$.

We have:

Thus $\phi$ is a homomorphism by definition.