= Degree-zero de Rham cohomology
{title2=$H^0_{\mathrm{dR}}(M)=\{f\in C^\infty(M):df=0\}$}
There are no exact forms in degree zero. A <smooth function> has zero <exterior derivative> precisely when it is <locally constant>, so $H^0_{\mathrm{dR}}(M)$ is the space of <locally constant functions>. On a connected nonempty <smooth manifold>, it is $\mathbb R$. An overlapping cover by two connected open subsets therefore does not give the direct-sum formula for <de Rham cohomology of a finite disjoint union>: restriction sends a constant to equal constants on both pieces.
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