Group Homomorphism/Examples/Pointwise Addition on Continuous Real Functions on Closed Unit Interval/Direct Sum

Example of Group Homomorphisms
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.

Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be a homomorphism from $\struct {\map {\mathscr C} J, +}$ to $\struct {\R, +}$.

Let $\phi$ satisfy the condition:
 * $\forall c \in \R: \map \phi {f_c} = c$

where $f_c$ is the constant mapping on $\R$ defined as:
 * $\forall x \in \R: \map {f_c} x = c$

Then $\struct {\map {\mathscr C} J, +}$ is the internal group direct product of $\map \ker \phi$ and the subgroup of constant mappings on $\R$.