ZANCO Journal of Pure and Applied Sciences The official scientific journal of Salahaddin University-Erbil https://zancojournals.su.edu.krd/index.php/JPAS ISSN (print ):2218-0230, ISSN (online): 2412-3986, DOI: http://dx.doi.org/10.21271/zjpas RESEARCH PAPER On Infinitesimal   -smooth Functions. Ibrahim O. Hamad Department of Mathematic, College of Science, Salahaddin University-Erbil, Kurdistan Region, Iraq A B S T R A C T: The aim of this paper is to study smoothness, approximate continuity, and approximate derivative in a nonstandard manner with respect to infinitesimal parameters. The new nonstandard introduced definitions are combined with standard and nonstandard intermediate value property. Particularly, we show that the existence     of continuous and smooth function has the infinitesimal intermediate value property. Moreover, for the same result, we reduce the continuity condition to the infinitesimal intermediate value condition KEY WORDS: infinitesimals, smooth functions, intermediate value property, continuity, symmetric functions. DOI: http://dx.doi.org/10.21271/ZJPAS.32.0.0 ZJPAS (2020) , 32(1);1-16 . 1.INTRODUCTION : 1.1. Nonstandard Analysis Background During 1961-1966, A. Robinson developed a first rigorous foundation of nonstandard analysis (NSA) (Robinson, 1961; Robinson, 1996). He had been using theorems in mathematical logic in the holies to derive known mathematical results in a non-classical way. His method based on the theory of models for making an extension of the classical number systems by looking at nonstandard models of their respective theories. Infinitely small and infinitely large numbers were to be found in the enlargement  of . He was able to justify proofs using infinitesimals and that was not possible before his discovery. His article with Bernstein certainly showed that these methods were able to produce original solutions to unsolved mathematical questions as well (Robert, 1988). Throughout this paper, by , we mean the proper extension of conventional real numbers , includes all real numbers together with nonstandard quantities. The elements of  are often called hyperreals. In 1977 Nelson, E (Nelson, 1977) introduce a new approach to constructing NSA, depending on three basic principles named; Transfer, Idealization, Standardization. His approach known by Internal Set Theory (IST). In IST, every mathematical object is regards to be a set, and every set is standard and any set or formula is called internal in case it does not defend with the new predicate "standard" and its derivations. There are a several different approaches has been presented by other mathematicians for representing a Robinson’s nonstandard analysis sense, one of the recent approach has been done by (Abdeljalil, 2018), he proposed a very simple method in practice to nonstandard analysis without using the ultrafilter. There were a number of studies have examined a results on nonstandard analysis and its application. A first nonstandard generalization of curvature and torsion and some other concepts in differential geometry had been presented by Hamad, I. (Hamad, Generalized curvature and torsion in nonstandard analysis, 2011). (Sun, 2015) were introduced an applications economics. Similarly