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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 ...
Covering nonstandard calculus, it develops differential and integral calculus using the hyperreal numbers, which include infinitesimal elements. These applications of nonstandard analysis depend on the existence of the standard part of a finite hyperreal r. The standard part of r, denoted st(r), is a standard real number infinitely close to r ...
In non-standard calculus the limit of a function is defined by: = if and only if for all , is infinitesimal whenever x − a is infinitesimal. Here R ∗ {\displaystyle \mathbb {R} ^{*}} are the hyperreal numbers and f* is the natural extension of f to the non-standard real numbers.
The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment.
In model theory, a discipline within mathematical logic, a non-standard model is a model of a theory that is not isomorphic to the intended model (or standard model). [1]
Traditionally, the variables dx and dy are considered to be very small (infinitesimal), and this interpretation is made rigorous in non-standard analysis. differentiation rules. direct comparison test A convergence test in which an infinite series or an improper integral is compared to one with known convergence properties. Dirichlet's test
2.10 Other standard special functions. ... is an example of continuous function that is nowhere differentiable; ... a non-profit organization.
In non-standard calculus, differentials are regarded as infinitesimals, ... For example, if f is a function of two variables x and y, then = ...
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