A Study on Some Fundamental Properties of Continuity and Differentiability of Functions of Soft Real Numbers
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.
Following the seminal work of Zadeh  on fuzzy set theory, development of mathematical theory and their applications in handling the problems under uncertain environment have been gaining momentum day by day. Considering some difficulties in the parametrization process in fuzzy set theory, Molodtsov in 1999  introduced an idea of soft set as a parametrized family of sets where parameter set takes values from an arbitrary set. He also showed the applications of soft sets in fields like smoothness of functions, probability theory, measure theory, and game theory.
After that Maji et al. [3, 4] defined some operations on soft sets based on which Shabir and Naz  introduced soft topologies, Aktas and Cagman  soft group, and Nazmul and Samanta  soft topological group. Recently Das et al. [8, 9] introduced the idea of soft metric and soft normed linear space. In  Das and Samanta also introduced the concept of soft real numbers. Using this concept they studied some basic properties of soft real numbers. However in their study on functions they have considered functions over crisp sets and have used extension principle for getting images of soft sets. In 2016  Tantawy and Hassan studied some basic operations like supremum and infimum in soft setting.
However, no studies have been found for functions from soft numbers to soft numbers. To deal with this type of functions is something different from the previous one. In this paper we have considered such type of functions and have studied some fundamental properties of continuous functions, like Bolzano’s property, intermediate value property, and fixed point property. The extension of Bolzano’s theorem in soft setting is crucial and from which some other properties, like intermediate value property and fixed point property, follow. We have also introduced the concept of differentiation of such functions and have extended Rolle’s theorem and Lagrange’s theorem in soft settings.
Unless otherwise stated all over this paper is taken as the parameter set and any soft set taken in this paper is assumed to be parameterwise nonempty.
Definition 1 (see  (soft set)). Let be a nonempty set and be a set which is called index set. A pair , where is a mapping, is called a soft set on .
Definition 2 (see  (soft element)). Let be a nonempty set and be a nonempty parameter set. Then a function is said to be a soft element of . A soft element of is said to belong to a soft set of , denoted by , if , . Thus a soft set of with respect to the index set can be expressed as .
Definition 3 (see ). Let be the set of real numbers, be the collection of all nonempty bounded subsets of , and be a set of parameters. Then a mapping is called a soft real set. It is denoted by .
If in particular is a singleton soft set, then, identifying with the corresponding soft element, it will be called a soft real number.
We use the notation to denote soft real numbers whereas will denote a particular type of soft real numbers such that , for all etc. Note that, in general, is not related to .
Definition 4 (see ). For two soft real numbers we define the following:(1) if , for all .(2) if , for all .(3) if , for all .(4) if , for all .
Definition 5 (see ). If are soft numbers, then modulus, sum, difference, product, and division of soft real numbers are denoted by , , , and , respectively, and defined by the following:(i), for all .(ii), for all .(iii), for all .(iv), for all .(v), for all .
Theorem 6 (see ). For any soft set , (where and , where is a set of soft real numbers).
Definition 7 (see ). Let ; then is said to be subset of and denoted by , if , .
Definition 8 (see  (equality of soft real sets)). Let ; then is said to be equal to and denoted by , if , .
Definition 9 (see ). The union of two soft sets and over the common universe is the soft set , where and for all We write .
Definition 10 (see ). The intersection of two soft sets and over the common universe is the soft set , where and for all . We write .
Definition 11 (see ). A real soft set is said to be bounded from above if there exists a soft real number such that , .
Definition 12 (see ). A real soft set is said to be bounded from below if there exists a soft real number such that , .
Definition 13 (see ). In a soft normed linear space a sequence of soft elements is said to be convergent and converges to a soft element if, for any soft real number , there exists a soft natural number such that , , and is denoted by or as , where is called the soft limit of the sequence .
Theorem 14. Let be a sequence in soft real set over a finite parameter set converging to ; then converge to uniformly over parameterwise; that is, for any there exists a (set of natural numbers) such that , .
Theorem 15 (see ). A sequence iff for all .
3. Elementary Set Theory
From this section all the study has done on soft real sets.
Definition 16. If are two soft real numbers with ; then is said to be soft closed interval with boundary points and . Clearly . So any closed soft interval can be taken as a soft set, where .
Definition 17. A length function of a soft interval is defined by length of the interval and denoted by .
Definition 18. A soft real set is said to be bounded if it is bounded from above and bounded from below.
Proposition 19. A soft real set is bounded iff are bounded sets in , .
Proof. We have a soft real set being bounded if there exists a soft real number such that for all iff for any , for all iff are bounded sets in , .
Definition 20. A soft number is said to be the least upper bound of a soft set if , , and for any there exists a such that .
Definition 21. A soft number is said to be greatest lower bound of a soft set if , , and for any there exists a such that .
Proposition 22. Every bounded soft real set has a least upper bound (l.u.b.) and a greatest lower bound (g.l.b.).
Proof. Let be a bounded soft real set. Since ’s are bounded set in , l.u.b. of exists; say . Take , . Now for , there exists such that , ; that is, where . The proof for greatest lower bound is similar.
Note. Uniqueness of l.u.b and g.l.b in a soft set follows from the uniqueness of l.u.b and g.l.b of crisp set.
Theorem 23 (nested interval theorem). If is a sequence of nested soft closed intervals satisfying the properties and as ; then is a singleton set .
Proof. Since for any , is a closed interval , by the definition of subset of soft sets is a sequence of sets satisfying the properties of nested intervals theorem. Hence , which implies (where ).
Definition 24. A soft set is said to be soft closed if for any , is closed set in .
Definition 25. Let be sequence of soft real numbers. Then is said to be a soft subsequence of if is a subsequence of for all .
Proposition 26 (Bolzano-Weierstrass’s form). Let be a sequence of soft real numbers in a bounded soft closed set . Then we can construct a soft subsequence of which is convergent in .
Proof. Let be a sequence of soft real numbers in a bounded soft closed set . Since the sequence is bounded and in closed set , by Bolzano-Weierstrass’s theorem there exists a subsequence converging to some point . Now if we construct a sequence of soft numbers such that is the th member of the subsequence , since for every sequence is convergent to a point , then for any there exists such that , , which holds for all . Hence , , where . Hence the sequence is convergent in .
Proposition 27 (Bolzano-Weierstrass’s theorem in finite parameter set). If is a sequence in a bounded soft closed set , then there exists a subsequence of converging to some .
Proof. Let be a sequence in soft bounded closed set and the parameter set . Then is a sequence in bounded closed set , so by Bolzano-Weierstrass’s theorem has a subsequence converging to some point in (say ). Next consider the subsequence in soft bounded closed set , and by the similar arguments has a subsequence converging to some point in (Say ). Continuing this process, we have a subsequence of , where converges to some point in (Say ). By the construction it is clear that is a subsequence of converging to for all . Hence converges to for all , which implies , where for all
4. Limit and Continuity
Definition 28 (function of soft sets). Let and be two soft real sets. Then is said to be a function of soft sets from to if sends a soft element of to a unique soft element of .
Definition 29. A soft set is said to be a soft domain if, for each , is an open set in .
Definition 30. A soft set is said to be a neighborhood of if is a neighborhood of , .
Definition 31. A function of soft sets is said to be constant in a domain if the function of soft sets , , where is a fixed soft element of .
Definition 32 (limit of a function of soft sets). Let be a function of soft sets and . Let a function of soft sets be defined in some deleted neighborhood of . Then is said to be the soft limit of the function of soft sets as tends to and is denoted by as or if for any there exists such that whenever Further if , then we call the function of soft sets continuous at .
Proposition 33. Soft limit of a function of soft sets is unique.
Proposition 34. Let be a function of soft sets and . Let a function of soft sets be defined in some deleted neighborhood of . If , then there exists soft deleted neighborhood of such that is bounded therein.
Proof. Since , so for there exists a such that whenever Thus, whenever , which shows the result.
Result 1. If , then for there exists such that whenever Therefore, . Thus, whenever .
Definition 35. A function of soft sets is said to be continuous in a soft set if the function of soft sets is continuous at any soft element .
Proposition 36 (limit theorem of functions of soft sets). Let and be two functions of soft sets defined in some soft domain . Let be a soft point of . Let and . Then,(1);(2);(3);(4);(5) if for all in some deleted neighborhood of . Proofs are exactly the same as the proofs of limit of crisp case.
Proposition 37. If is a function of soft sets continuous at and (), then there exists such that () satisfying .
Proof. Let . Then . By the continuity of , there exists a such that whenever ; that is, whenever ; that is, whenever . The argument for is similar.
Corollary 38. If is a function of soft sets continuous at and (), then there exists such that () satisfying .
Proof. The proof directly follows from Proposition 37 by taking where .
Proposition 39. Let be a continuous function of soft sets on . If uniformly over parameter in , then .
Proof. Let be continuous at . Now for any there exists a such that whenever . Since , there exists a soft natural number such that , . Hence , . Thus, .
Lemma 40. If and , , then .
Proof. Let , . If for any , , then for there exists a soft natural number such that , , , which is a contradiction. Hence , ; that is, .
We are now in a position to consider the extension of Bolzano’s theorem in soft setting. This result plays a crucial role for several other theorems to follow immediately. The proof of this theorem is interesting.
Proposition 41 (Bolzano’s form). If a function of soft sets is continuous on and , then for any there exists such that .
Proof. Without loss of generality, let . Let us construct a set . Since is continuous at , there exists a such that for all . Therefore, the set is nonempty. Clearly the soft real numbers form a poset under the relation on . So by Hausdorff maximality principle there exists a maximal totally ordered subset of . Let for all . Since , . Hence clearly . Now if , then the theorem is proved. If not, then either (i) or (ii) .
Since s are the members of the totally ordered set , for any , or . Hence , .
Now we shall show that
Proof. Since and for all , by the property of supremum for any , there exists a such that ; that is, , which implies . Since is arbitrary, . Hence .
Now consider the possibility (i) If . Then by the continuity of at there exists a such that for every in . Therefore, we can find a with in such that , which contradicts the . Hence possibility (i) is not possible; that is, cannot be greater than 0.
Next consider the possibility (ii) . We show that either there exists a point to serve the theorem or . By for any in , . Choose any with for some and for some . If for any such soft number, then the theorem is proved. If , then proceeding as in possibility (i) we have a contradiction. So if for such soft numbers, then ; that is, implies . Thus, either there exists a soft number to serve the theorem or .
Now if and since (as and , which follows from the condition and ), then there exists a such that . Choose a such that . Now if we define as for and , then ; otherwise, it contradicts that is the supremum of the maximal chain . Therefore, there exists a such that . If again, the theorem is served. If , then, since but , by construction of it follows that and , . Now construct another soft number such that for and . Clearly , so as we are considering the remaining possibility .
Now and and . Take . If , then the theorem is proved. If choose and ; otherwise and . In the similar argument choose , and if , then the theorem is proved. If choose and ; otherwise and . In this way if , then the theorem is proved. Otherwise we get sequences and such that and . Clearly the sequence , where for all , satisfies the nested intervals theorem. Therefore, there exists a unique and , as . By the construction of the sequence , for all . Hence and as , where for all and . Since the sequences only vary in the th parameter and are constant in the other parameter, the sequences converge to parameterwise uniformly. Consequently, and as , but and for all , which shows that and . Hence .
Corollary 42. If a function of soft sets is continuous on and if for , , then there exists such that .
Remark 43. If a function of soft sets is continuous on and , then there may not exist any such that , which can be shown by the following example.
Example 1. Let be a function of soft sets with parameter set defined on by and . Clearly is continuous on , but only when , which does not belong to the domain set, hence showing the result.
Proposition 44. If is a continuous function on a soft interval over a finite parameter set , then is bounded.
Proof. If possible let be not bounded above. Then there does not exist any such that ; that is, there exists a , such that for every there exists a sequence with (set of natural numbers). Now since is sequence in , by Proposition 27, there exists a subsequence of converging to some soft number . Since the set parameter is finite, so the sequence uniformly over parameter, which implies . However, , , which implies , , a contradiction. Hence is bounded.
Note. If the parameter set is not finite then a continuous function on a soft interval may not be bounded, which will considered by Example 2.
Lemma 45. Let be the closed interval in endowed with the usual topology. If , then there exists an unbounded continuous function with respect to box topology in .
Proof. Clearly is a normal space and is a closed subset of . Now define a function such that if , , if is the first nonzero in the sequence , and if is the first nonzero in the sequence . Since, with the box topology, the subspace topology on is discrete topology, is continuous and by construction is unbounded in . Now since is a normal space, by Tietze’s extension theorem there exists a continuous function defined on such that , , so is unbounded, hence the lemma.
Example 2. Let us consider the soft interval and . If we define a function of soft sets such that if and if , where is as in Lemma 45, taking any , then since is continuous, for any there exists , such that whenever , , which shows that whenever (where for ) and . Hence is continuous but unbounded.
Proposition 46 (intermediate value property). If a function of soft real sets is continuous on and if for some , then there exists between and such that .
Proof. Consider the function of soft sets . Then clearly is continuous in and . Therefore, by Corollary 42 there exists such that ; that is, ; that is, .
Proposition 47 (fixed point theorem). If a function of soft real sets is continuous on and the value is also in the soft interval , then for any there exists such that .
Proof. Consider the function of soft sets . If or , then the proof is over. If not, then clearly is continuous in and . Thus, by Corollary 42 there exists such that ; that is, ; that is, .
Result 2. Let be a continuous function of soft sets on . Further if is bounded, there may not exist any soft real number in attaining the bound parameterwise.
Consider the function as in Lemma 45 and in . Clearly the range set of is , so is continuous. Now if we consider the function as in Example 2, taking in place of , then exactly by the similar argument is a continuous function of soft sets in with range set in parameter being by Proposition 46 (intermediate value property), which shows the result.
Proposition 48. Let a function of soft sets be continuous on a soft set and be a real number. Ifthen the function of soft sets is continuous.
Proof. Let . Since is continuous for any , there exists such that whenever . By the definition of we have . Hence is continuous.
Proposition 49. Let a function of soft sets be continuous on a soft set . If then the function of soft sets is continuous.
Proof. We have Since is continuous at , there exists a such that whenever (by result 4.8), which implies . Hence for any , , and for , can be made less than directly by the continuity of .
Remark 50. Let a function of soft sets from to satisfy the condition that for any , implies . Then if we define for any if , then clearly is a function from to .
Proposition 51. A function of soft sets satisfying the property of Remark 50 is continuous at iff the function is continuous at , .
Proof. Let the function of soft sets be continuous at . Take any ; then for , there exists a soft element such that whenever ; that is, whenever , ; that is, whenever (since is independent for the other parameter), which shows that is continuous at , .
Conversely, let be continuous at , . Take any . Since is continuous at , , so for every , there exists such that whenever . Now if we take , , then we have whenever