Derivations of smooth functions are vector fields (${\mathrm{Der}}_{\mathbb{R}}(C^{\infty }(M))\cong \Gamma (TM)$)

Every real-linear derivation of an algebra $D:C^{\infty }(M)\to C^{\infty }(M)$ on a smooth manifold has the form $Dh=Z(h)$ for a unique smooth vector field $Z$. The product rule makes $D$ local: multiplying a function that vanishes near a point by a smooth cutoff function supported there gives zero derivative at that point. Locally write $h(x)=h(p)+{\sum }_i(x^i-x^i(p))h_i(x)$, where $h_i(p)={\partial }_{i}h(p)$. Applying $D$ at $p$ gives $Dh(p)={\sum }_{i}D(x^i)(p){\partial }_{i}h(p)$; the smooth coefficients $D(x^i)$ define the claimed vector field. No continuity hypothesis is needed. In dimension zero all such derivations vanish.