| The theory of fuzzy mathematics is a new branch of mathematics. Since the theorywas first introduced in 1960s, it displays powerful spirit and widely development. Afterseveral years'developing, the theory has many branches. The theory of fuzzy analysis,which is an extension of classical analysis, is an important branch of the theory of fuzzymathematics. Fuzzy analysis has been applied in many areas. The study of the spaceof fuzzy numbers is an indispensable part of fuzzy analysis. Therefore, the study of thespace of fuzzy numbers has great value of theory and application. This paper is inspiredby the phenomenon of diffusion. Since the boundary of diffusion of gases and liquids isnot crisp, it is difficult to measure area and size of the diffusion. Hence, fuzzy numbersmight be a way to represent the length, width and height of the diffusion. The integral anddifferential over fuzzy directed line are studied in this paper to measure the area of waterpolluted by oil leaking out of oil-tanker. Secondly, the convergence of fuzzy numbersendowed with sendograph metric is discussed. Then, we discuss the subdifferential ofconvex fuzzy mapping. At last, the fuzzy mappings extended by continuous functions arestudied. Our main work is as follows:1. We give the definition of fuzzy directed line and show several its properties. Thenwe define the fuzzy integral, which is the extension of fuzzy integral with real variables,over the fuzzy directed line. The definitions of fuzzy quotient and fuzzy differential areshown. We also obtain a theorem which is similar to the fundamental theorem of classicalcalculus.2. After studying the properties of sendograph metric, we show two attainabilitytheorems, i.e. the sendograph metric of two fuzzy numbers can be determined by maxi-mum of the distances between two points and the sendograph of those two fuzzy numbersrespectively. The two points are in special subsets of sendograph of those fuzzy numbers.Then we obtain Monotone convergence theorem and Nested theorem of intervals in thespace of fuzzy numbers endowed with sendograph metric3. The definition of convex fuzzy mapping and its fuzzy subdifferential are shown.After studying the properties of fuzzy subdifferential, we obtain the necessity and suffi-ciency condition of which the solution of convex fuzzy programming is minimum solu- tion. Then we show the application of fuzzy subdifferential in convex fuzzy programming.4. After studying the fuzzy mappings which are extended by real continuous func-tions according to Zadeh extension principle, we obtain that the values of these fuzzymapping on fuzzy numbers are also fuzzy numbers. Naturally, the algebra operations ofany two fuzzy numbers are also fuzzy numbers. From which we show that the proof ofProposition 3.5 in Kim's paper is incorrect. At same time, a new proof of the Propositionis shown.
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