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Advances in Fuzzy Systems
Volume 2018 (2018), Article ID 6429572, 8 pages
https://doi.org/10.1155/2018/6429572
Research Article

A Study on Some Fundamental Properties of Continuity and Differentiability of Functions of Soft Real Numbers

1Department of Mathematics, Bidhannagar College, Salt Lake, Sector I, Kolkata, West Bengal 700064, India
2Department of Mathematics, Visva Bharati, Santiniketan, West Bengal 731235, India

Correspondence should be addressed to Ramkrishna Thakur; moc.liamg@60marrukaht

Received 15 November 2017; Accepted 21 December 2017; Published 1 March 2018

Academic Editor: Katsuhiro Honda

Copyright © 2018 Ramkrishna Thakur and S. K. Samanta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a new type of functions from a soft set to a soft set and study their properties under soft real number setting. Firstly, we investigate some properties of soft real sets. Considering the partial order relation of soft real numbers, we introduce concept of soft intervals. Boundedness of soft real sets is defined, and the celebrated theorems like nested intervals theorem and Bolzano-Weierstrass theorem are extended in this setting. Next, we introduce the concepts of limit, continuity, and differentiability of functions of soft sets. It has been possible for us to study some fundamental results of continuity of functions of soft sets such as Bolzano’s theorem, intermediate value property, and fixed point theorem. Because the soft real numbers are not linearly ordered, several twists in the arguments are required for proving those results. In the context of differentiability of functions, some basic theorems like Rolle’s theorem and Lagrange’s mean value theorem are also extended in soft setting.