Standard part function

In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal ${\displaystyle x}$, the unique real ${\displaystyle x_{0}}$ infinitely close to it, i.e. ${\displaystyle x-x_{0}}$ is infinitesimal. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat,^[1] as well as Leibniz's Transcendental law of homogeneity.

The standard part function was first defined by Abraham Robinson who used the notation ${\displaystyle {}^{\circ }x}$ for the standard part of a hyperreal ${\displaystyle x}$ (see Robinson 1974). This concept plays a key role in defining the concepts of the calculus, such as continuity, the derivative, and the integral, in nonstandard analysis. The latter theory is a rigorous formalization of calculations with infinitesimals. The standard part of x is sometimes referred to as its shadow.^[2]