In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic.

Non-rigorous calculations with infinitesimals were widely used before Karl Weierstrass sought to replace them with the (ε, δ)-definition of limit starting in the 1870s. (See history of calculus.) For almost one hundred years thereafter, mathematicians such as Richard Courant viewed infinitesimals as being naive and vague or meaningless.^[1]

Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt and Jerzy Łoś. According to Howard Keisler, "Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century."^[2]

The history of nonstandard calculus began with the use of infinitely small quantities, called infinitesimals in calculus. The use of infinitesimals can be found in the foundations of calculus independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s. John Wallis refined earlier techniques of indivisibles of Cavalieri and others by exploiting an infinitesimal quantity he denoted ${\displaystyle {\tfrac {1}{\infty }}}$ in area calculations, preparing the ground for integral calculus.^[3] They drew on the work of such mathematicians as Pierre de Fermat, Isaac Barrow and René Descartes.

In early calculus the use of infinitesimal quantities was criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley in his book The Analyst.

Several mathematicians, including Maclaurin and d'Alembert, advocated the use of limits. Augustin Louis Cauchy developed a versatile spectrum of foundational approaches, including a definition of continuity in terms of infinitesimals and a (somewhat imprecise) prototype of an ε, δ argument in working with differentiation. Karl Weierstrass formalized the concept of limit in the context of a (real) number system without infinitesimals. Following the work of Weierstrass, it eventually became common to base calculus on ε, δ arguments instead of infinitesimals.

This approach formalized by Weierstrass came to be known as the standard calculus. After many years of the infinitesimal approach to calculus having fallen into disuse other than as an introductory pedagogical tool, use of infinitesimal quantities was finally given a rigorous foundation by Abraham Robinson in the 1960s. Robinson's approach is called nonstandard analysis to distinguish it from the standard use of limits. This approach used technical machinery from mathematical logic to create a theory of hyperreal numbers that interpret infinitesimals in a manner that allows a Leibniz-like development of the usual rules of calculus. An alternative approach, developed by Edward Nelson, finds infinitesimals on the ordinary real line itself, and involves a modification of the foundational setting by extending ZFC through the introduction of a new unary predicate "standard".

To calculate the derivative ${\displaystyle f'}$ of the function ${\displaystyle y=f(x)=x^{2}}$ at x, both approaches agree on the algebraic manipulations:

${\displaystyle {\frac {\Delta y}{\Delta x}}={\frac {(x+\Delta x)^{2}-x^{2}}{\Delta x}}={\frac {2x\Delta x+(\Delta x)^{2}}{\Delta x}}=2x+\Delta x\approx 2x}$

This becomes a computation of the derivatives using the hyperreals if ${\displaystyle \Delta x}$ is interpreted as an infinitesimal and the symbol "${\displaystyle \approx }$" is the relation "is infinitely close to".

In order to make f ' a real-valued function, the final term ${\displaystyle \Delta x}$ is dispensed with. In the standard approach using only real numbers, that is done by taking the limit as ${\displaystyle \Delta x}$ tends to zero. In the hyperreal approach, the quantity ${\displaystyle \Delta x}$ is taken to be an infinitesimal, a nonzero number that is closer to 0 than to any nonzero real. The manipulations displayed above then show that ${\displaystyle \Delta y/\Delta x}$ is infinitely close to 2x, so the derivative of f at x is then 2x.

Discarding the "error term" is accomplished by an application of the standard part function. Dispensing with infinitesimal error terms was historically considered paradoxical by some writers, most notably George Berkeley.

Once the hyperreal number system (an infinitesimal-enriched continuum) is in place, one has successfully incorporated a large part of the technical difficulties at the foundational level. Thus, the epsilon, delta techniques that some believe to be the essence of analysis can be implemented once and for all at the foundational level, and the students needn't be "dressed to perform multiple-quantifier logical stunts on pretense of being taught infinitesimal calculus", to quote a recent study.^[4] More specifically, the basic concepts of calculus such as continuity, derivative, and integral can be defined using infinitesimals without reference to epsilon, delta (see next section).

Keisler's Elementary Calculus: An Infinitesimal Approach defines continuity on page 125 in terms of infinitesimals, to the exclusion of epsilon, delta methods. The derivative is defined on page 45 using infinitesimals rather than an epsilon-delta approach. The integral is defined on page 183 in terms of infinitesimals. Epsilon, delta definitions are introduced on page 282.

The hyperreals can be constructed in the framework of Zermelo–Fraenkel set theory, the standard axiomatisation of set theory used elsewhere in mathematics. To give an intuitive idea for the hyperreal approach, note that, naively speaking, nonstandard analysis postulates the existence of positive numbers ε which are infinitely small, meaning that ε is smaller than any standard positive real, yet greater than zero. Every real number x is surrounded by an infinitesimal "cloud" of hyperreal numbers infinitely close to it. To define the derivative of f at a standard real number x in this approach, one no longer needs an infinite limiting process as in standard calculus. Instead, one sets

${\displaystyle f'(x)=\mathrm {st} \left({\frac {f^{*}(x+\varepsilon )-f^{*}(x)}{\varepsilon }}\right),}$

where st is the standard part function, yielding the real number infinitely close to the hyperreal argument of st, and ${\displaystyle f^{*}}$ is the natural extension of ${\displaystyle f}$ to the hyperreals.

A real function f is continuous at a standard real number x if for every hyperreal x' infinitely close to x, the value f(x' ) is also infinitely close to f(x). This captures Cauchy's definition of continuity as presented in his 1821 textbook Cours d'Analyse, p. 34.

Here to be precise, f would have to be replaced by its natural hyperreal extension usually denoted f^* (see discussion of Transfer principle in main article at nonstandard analysis).

Using the notation ${\displaystyle \approx }$ for the relation of being infinitely close as above, the definition can be extended to arbitrary (standard or nonstandard) points as follows:

A function f is microcontinuous at x if whenever ${\displaystyle x'\approx x}$, one has ${\displaystyle f^{*}(x')\approx f^{*}(x)}$

Here the point x' is assumed to be in the domain of (the natural extension of) f.

The above requires fewer quantifiers than the (ε, δ)-definition familiar from standard elementary calculus:

f is continuous at x if for every ε > 0, there exists a δ > 0 such that for every x' , whenever |x − x' | < δ, one has |f(x) − f(x' )| < ε.

A function f on an interval I is uniformly continuous if its natural extension f* in I* has the following property (see Keisler, Foundations of Infinitesimal Calculus ('07), p. 45):

for every pair of hyperreals x and y in I*, if ${\displaystyle x\approx y}$ then ${\displaystyle f^{*}(x)\approx f^{*}(y)}$.

In terms of microcontinuity defined in the previous section, this can be stated as follows: a real function is uniformly continuous if its natural extension f* is microcontinuous at every point of the domain of f*.

This definition has a reduced quantifier complexity when compared with the standard (ε, δ)-definition. Namely, the epsilon-delta definition of uniform continuity requires four quantifiers, while the infinitesimal definition requires only two quantifiers. It has the same quantifier complexity as the definition of uniform continuity in terms of sequences in standard calculus, which however is not expressible in the first-order language of the real numbers.

The hyperreal definition can be illustrated by the following three examples.

Example 1: a function f is uniformly continuous on the semi-open interval (0,1, if and only if its natural extension f* is microcontinuous (in the sense of the formula above) at every positive infinitesimal, in addition to continuity at the standard points of the interval.

Example 2: a function f is uniformly continuous on the semi-open interval edit