Group Homomorphism/Examples/Pointwise Addition on Continuous Real Functions on Closed Unit Interval/Example 3/Kernel

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

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

Let $I_J$ denote the identity mapping on $J$:
 * $\forall x \in J: \map {I_J} x = x$

Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be the homomorphism defined as:
 * $\forall f \in \map {\mathscr C} J: \map \phi f = \ds \int_0^1 \map f x \rd x$

The kernel of $\phi$ is given by:
 * $\map \ker \phi = I_J - f_m$

where:
 * $f_m: \R \to \R$ denotes the constant mapping on $\R$
 * $m = \dfrac 1 2$
 * $I_J$ denotes the identity mapping on $J$.

Proof
From Group Homomorphism: Example 3, we have that $\phi$ is indeed a homomorphism.

For all $c \in \R$, let $f_c: \R \to \R$ be the constant mapping:
 * $\forall x \in \R: \map {f_c} x = c$

First we show that:
 * $\forall c \in \R: \map \phi {f_c} = c$

Let $c \in \R$ be arbitrary.

We have:

Then we show that there exists a unique $m \in \R$ such that:
 * $\map \phi {I_J - f_m} = 0$

where in this case:
 * $m = \dfrac 1 2$

We have:

Hence the result by definition of kernel.