I am reading Color for the Sciences by Jan Koenderink, and in Ch. 3 he introduces the dual number system to define the space of possible power spectra for a beam of light. However, his statements do not seem mathematically rigorous and I cannot see a way they could be made sensible.
He appears to be assuming there is a natural way to extend functions defined on the reals to the duals, but this post suggests this is impossible.
I see how the "blip function" he defines is ruled out. However, this is not the only way to extend the blip function on $\mathbb R$ to the duals, and it seems one could simply define a blip function on the duals that agrees with the one on $\mathbb R$ and also with his criterion for acceptable functions.
Finally, his claim at the end that the space is Hausdorff implies that there exists a topology, which he has not specified. Perhaps there is a natural topology on this space that is too obvious to state?
Formally, one can see the set of dual numbers as the quotient $\Bbb R[t]/(t^2)$ and write $\epsilon$ for the class of the indeterminate $t$ in the quotient. One equips the set of dual numbers with the Euclidean topology. No extension of the blip function to the set of dual numbers is even continuous, because the restriction to the real axis is already not continuous. His space $\Bbb S$ is an infinite dimensional subspace of $\mathcal{C}^\infty(\Bbb R)$ --- which is a Frechét space equipped with the family of semi-norms $\|f\|_{k,n} = \sup_{x \in [-n,n]}|f^{(k)}(x)|$ --- because this is his assumption: he just restricts the discussion to smooth functions.
From what I see, the author is being mathematically sound and not really sloppy here.