Standard part function

Remarks

Existence: $\operatorname{st}(x)$ is the supremum of the set $\set{q \in \R : q < x}$, which is nonempty and bounded above when $x$ is finite. Uniqueness: if $x - r$ and $x - r'$ are infinitesimal for reals $r$ and $r'$, then $r - r'$ is a real infinitesimal, so $r = r'$.

Example: for a positive infinitesimal $\varepsilon$, $\operatorname{st}(3 + \varepsilon) = 3$, since $(3 + \varepsilon) - 3 = \varepsilon$ is infinitesimal.

Non-example: $H = 1/\varepsilon$ is not finite, and for every real $r$ the difference $H - r$ is larger than $1$, so it is not infinitesimal and $H$ has no standard part.

The finite hyperreals form a subring of ${}^*\R$, and $\operatorname{st}$ is a ring homomorphism onto $\R$ whose kernel is the set of infinitesimals. It is also monotone: $x \le y$ implies $\operatorname{st}(x) \le \operatorname{st}(y)$.