# Derivations on the continuous functions of a manifold

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.

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 $$M$$.