Approximately Continuous Transformations

Approximately Continuous Transformations

1. An interesting class of real functions of a single real variable, the approximately continuous functions, was introduced by Denjoy [l] in his work on derivatives. The two principal facts discovered by Denjoy are that these functions are of Baire class 1 and have the Darboux property. Ridder [2] showed that the arguments of Denjoy apply to real functions of n variables. In this paper we discu...

Continuity of super- and sub-additive transformations of continuous functions

We prove a continuity inheritance property for super- and sub-additive transformations of non-negative continuous multivariate functions defined on the domain of all non-negative points and vanishing at the origin. As a corollary of this result we obtain that super- and sub-additive transformations of continuous aggregation functions are again continuous aggregation functions.

On Absolutely Continuous Transformations

Some Remarks on Continuous Transformations

Suppose / is a single-valued transformation of the metric space M onto the metric space fM. Many of the theorems of topology assert that if / is continuous, then it must have various other properties also. We deal here with the problem of determining under what circumstances certain of these properties are actually equivalent to continuity. The properties considered are: (1) fX is compact for e...

Transformations for semi-continuous data

Semi-continuous data arise in many applications where naturally-continuous data become contaminated by the data generating mechanism. The resulting data contain several values that are “too frequent”, and in that sense are a hybrid between discrete and continuous data. The main problem is that standard statistical methods, which are geared towards continuous or discrete data, cannot be applied ...