Derivations on algebras of one-point compactification of affine semigroups

For any affine semigroup $S$ the set $S\cup\{\infty\}$ has a natural structure of semigroup, additionally if $S$ is endowed with the discrete topology, the semigroup $S\cup\{\infty\}$ can be studied as the one-point compactification of $S$. In this article we study the derivations on the semigroup algebra $\mathbb{C}[S\cup\{\infty\}]$ in relation to the derivations on the semigroup algebra $\mathbb{C}[S]$ considering the metrizable topology on $\mathbb{C}[S\cup\{\infty\}]$ induced by the one-point compactification topology of $S\cup\{\infty\}$.