Continuity for s-convex fuzzy processes Juliana Cervelati 1 , Mar´ ıa Dolores Jim´ enez-Gamero 2 , Filidor Vilca-Labra 3 , and Marko Antonio Rojas-Medar 1 1 Departamento de Matem´atica Aplicada, Universidade Estadual de Campinas, CP 6065, 13083-970, Campinas-SP, Brazil ju@ime.unicamp.br, marko@ime.unicamp.br 2 Departamento de Estad´ ıstica e Investigaci´on Operativa, Universidad de Sevilla, 41012 Sevilla, Spain dolores@us.es 3 Departamento de Estat´ ıstica, Universidade Estadual de Campinas, CP 6065, 13083-970, Campinas-SP, Brazil fily@ime.unicamp.br In a previous paper we introduced the concept of s-convex fuzzy mapping and established some properties. In this work we study the continuity for s-convex fuzzy processes. 1 Introduction The notion of convex process was introduced by Rockafellar [19] (see also [20]). These processes are set-valued maps whose graphs are closed convex cones. They can be seen as the set-valued version of a continuous linear operator. Derivatives of some set-valued maps are closed convex processes, which is a desirable property for a derivative (see [1]). An important property of convex processes is that it is possible to transpose closed convex processes and to use the benefits of duality theory. As it is well known, these facts are very useful in optimization theory (see for example [2], [16], [17], [18], [3]). The extension of this notion to the fuzzy framework was done by Matloka [15]. Recently, Syau, Low and Wu [26] observed that Matloka’s definition is very strict. They gave another definition that extends Matloka’s one. The concept of m-convex fuzzy mapping was introduced in [7]. When m = 1 this concept and the definition of convex fuzzy process given in [26] coincide (see Theorem 3.4, p. 195 in [26]). As a generalization of convex functions, Breckner [4] introduced s- convex functions and in [5] he studied the set-valued version of these functions. Convex processes are a particular case of s-convex set-valued maps. Breckner also proved the important fact that a set-valued map is s-convex if and only if its support function is a s-convex function. Other related works are [6], [24], [25]. The fuzzy version of Breckner’s definition was introduced in [8], that was