Fuzzification of the theory of continuous functions

Abstract There are classical results in mathematics which are incomplete. An example is given by the well-known result of the classical mathematical analysis stating that a continuous function defined on a closed interval is bounded. Students study this theorem at the beginning of their courses of higher mathematics. But if we ask whether the converse is true we reveal that the answer is negative. The criterion of boundedness may be found only in terms of fuzzy set theory. The main result of this paper (Theorem 2) states that a function ƒ defined on a closed interval is bounded if and only if ƒ is fuzzy continuous. The local form of this result (Theorem 1) is also proved in the third section of the paper. In order to do this in the second section of the paper (after the introduction) the main concepts and constructions of the theory of fuzzy continuous functions are introduced and discussed. The third section is devoted to the problems of boundedness. In the fourth section operations on sets of fuzzy continuous functions are investigated.