Fixed point for derivative and differentiation of single-valued and set-valued functions on metric spaces

Study of the fixed point for derivative functions is an effort to expand the knowledge of fixed point for functions. This study represents original research on the existence of the fixed point for derivative functions which has been not studied before. Therefore this study attempts to explore the existence of fixed point for derivative functions. The research found that the derivative function defined on a closed unit interval into itself has a fixed point. In addition, this study attempts to extend those results for the derivative function defined on the whole real number line. By the concepts of commutativity and compatibility between the function and its derivatives show that the derivative function of the real-valued function has a fixed point. Meanwhile, in the case of set-valued function, we use the definition of the generalizations of the Hukuhara derivative. By using hybrid composite mapping compatible with Hausdorff metric, this study shows that derivative of the interval-valued function has a fixed point. Furthermore, based on the absolute derivative notion on metric spaces in the study of differentiation for single-valued functions, we introduce the new notions of the "Straddle Lemma" and the class of the "Darboux function". Other results in this study are the absolute derivative and the metric derivative of the set-valued functions. This expansion adds the literature on differentiability references for setvalued functions, among others the continuity of the set-valued function, absolute derivative of the constant set-valued function, and comparisons with the Hukuhara derivative and generalization of the Hukuhara derivative. The metric derivative concept introduced for the set-valued function generates the generalization of the famous Rademacher’s theorems