Let $R$ be a ring and $M$ be a manifold. For every $n$ define $$S_c^n(M;R)=\{\phi\in S^n(M;R)\vert \ \exists K\subseteq M \text{ compact}: \forall \psi:\Delta^n\rightarrow M \text{ with } \operatorname{im}(\psi)\cap K=\emptyset \text{ we have } \phi(\psi)=0\}.$$ Restricting the differential of singular cohomology makes $S_c^{\ast}(M;R)$ a cochain complex. Its cohomology groups are called compactly supported cohomology groups.
I read that the zeroth cohomology group with compact support (and coefficients in $R$) is the set of functions $M\rightarrow R$ with compact support that are constant on any continuous path. Is that true? Why is that?