The main goal of this paper is to characterize the family of all functions f which satisfy the following condition: whenever g is a Darboux function and f < g on ℝ there is a Darboux function h such that f < h < g on ℝ.
The classical Denjoy-Young-Saks theorem gives a relation, here termed the Denjoy property, between the Dini derivatives of an arbitrary one-variable function that holds almost everywhere. Concerning the possible generalizations to higher dimensions, A. S. Besicovitch proved the following: there exists a continuous function of two variables such that at each point of a set of positive measure there exist continuum many directions, in each of which one Dini derivative is infinite and the other...
Si dimostra che il funzionale è semicontinuo inferiormente su , rispetto alla topologia indotta da , qualora l’integrando sia una funzione non-negativa, misurabile in , convessa in , limitata nell’intorno dei punti del tipo , e tale che la funzione sia semicontinua inferiormente su .
Motivated by the study of multidimensional control problems of Dieudonné-Rashevsky type, we raise the question how to understand to notion of quasiconvexity for a continuous function f with a convex body K instead of the whole space as the range of definition. In the present paper, we trace the consequences of an infinite extension of f outside K, and thus study quasiconvex functions which are allowed to take the value +∞. As an appropriate envelope, we introduce and investigate the lower semicontinuous...