Some background to the seemingly vague question. I am studying some differential geometry and came across the notion of infinitesimal rotations . Now, the sense in which these infinitesimals are treated is rather unconventional (traditional, some would prefer), in that they are strictly non-zero, but are nilpotent --- that is, if $\varepsilon$ is some infinitesimal quantity, $\varepsilon\neq0$ and $\varepsilon^2=0$. There are some elementary papers in differential geometry that also handle infinitesimals in this manner. Though this is odd, it has been proved that such quantities exist in a plausible number system (not true if we are working within ZFC). Now on to the question.
Is it possible to define a function $f:R\to(-\varepsilon,\varepsilon)\subset R^*$ ($R^*$ being some number system involving the reals that allows infinitesimals to exist) such that each real $r$ is mapped to an infinitesimal quantity? If yes, then how can we rigorously define notions of continuity and differentiability? Will these mimic the conventional definitions, or do we need to treat these functions specially? Any help is appreciated. Thanks in advance!