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

Examples 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.

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