Functions on the zeroes of dx
  In the model of synthetic differential geometry consisting of sheaves (with respect to open covers) over the opposite category of the category of closed finitely generated C-infinity 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. 
James J. Faran V