A theorem on semi-continuous functions

A Theorem on Semi-Continuous Functions

A Theorem on Semi - Continuous Functions

RECENTLY G. C. Young* and A. Denjoyf have communicated theorems—those in Denjoy's memoir are of an especially comprehensive character—dealing, in particular, with point sets where the four derivatives of a given continuous function are identical. It is the purpose of this note to treat an analogous problem that arises when "derivative" is replaced by "saltus."$ However, instead of confining our...

On Differences of Semi-continuous Functions

Extrinsic and intrinsic characterizations are given for the class DSC(K) of differences of semi-continuous functions on a Polish space K, and also decomposition characterizations of DSC(K) and the class PS(K) of pointwise stabilizing functions on K are obtained in terms of behavior restricted to ambiguous sets. The main, extrinsic characterization is given in terms of behavior restricted to som...

A Theorem on Continuous Functions in Abstract Spaces

In this note the following Theorem A concerning continuous functions in a very general abstract space is established, and from this theorem are deduced certain results concerning semi-metric spaces. In particular, Theorems 2.2 and 2.3 below generalize a theorem proved by Montgomery concerning the behavior of the distances between points of a metric space under transformations of the space into ...

Computability on Continuou, Lower Semi-continuous and Upper Semi-continuous Real Functions

In this paper we investigate continuous and upper and lower semi-continuous real functions in the framework of TTE, Type-2 Theory of EEectivity. First some basic facts about TTE are summarized. For each of the function spaces, we introduce several natural representations based on diierent intiuitive concepts of \eeectivity" and prove their equivalence. Computability of several operations on the...