Let $G$ be the topological semigroup whose underlying space is $C(\mathbb{R}^d,\mathbb{R}^d)$ equipped with composition as semigroup operation and let $H$ be the topological group whose underlying space is $C(\mathbb{R}^d,\mathbb{R}^d)$ equipped with pointwise addition as group law; here $C(\mathbb{R}^d,\mathbb{R}^d)$ is equipped with the compact-open topology.


Is there a continuous (non-constant) semigroup homomorphism from $G$ to $H$?



Continuous semigroup homomorphism of composition to additive structure
The new version, for semigroups, is much easier: there is no such homomorphism $\varphi$, for purely algebraic reasons. Consider just the constant functions $c$. Since $cd=c$ in the semigroup, you must map $\varphi(d)=0$. But for any $f$, $fc$ is also constant ($=f(c)$), so $\varphi(f)+\varphi(c)=0$ and thus $\varphi(f)=0$ as well.


