Functions on the zeroes of dx

In the model ℱ of synthetic differential geometry consisting of sheaves (with respect to open covers) over F, the opposite category of the category of closed finitely generated C^∞-rings, any morphism from S, the zeroes of the "amazing right adjoint" of dx, to the real line R extends to a morphism from R to R. This shows that the De Rham cohomology of the space S is the same as the characteristic cohomology of the ideal generated by dx.