TITLE: Derivations on the continuous functions of a manifold
QUESTION [6 upvotes]: For a manifold $M$ a vector field is a derivation of the algebra $C^{\infty}(M)$ of smooth functions on $M$. What happens if look instead as derivations on the continuous functions of a manifold. I guess we get fewer derivations . . .  but I'm not sure how one might prove this.

REPLY [10 votes]: More is true: if $X$ is a topological manifold, then in fact $\operatorname{Der}(C(X)) = 0$, where $C(X)$ denotes the $\mathbb{R}$-algebra of $\mathbb{R}$-valued continuous functions on $X$. In particular, this is so for smooth manifolds $M$.
Here is one proof: https://ncatlab.org/nlab/show/derivation#DerOfContFuncts