= Derivations of smooth functions are vector fields
{title2=$\operatorname{Der}_{\mathbb R}(C^\infty(M))\cong\Gamma(TM)$}
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.
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