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 $H$?

invertiblefunctions on $\mathbb R^d$? $\endgroup$ – Christian Remling Nov 16 at 20:14