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, 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.