Onypa D. Cluster sets and oscillations

The thesis are devoted to the study of the problem of the construction of a function with a given characteristic such as a discontinuity point set, an oscillation, a limit oscillation or a cluster set. It is obtained that for a separable F_sigma-set of the first category in Eberlein space there is a quasi-continuous function, defined on that space, such that its set of discontinuities is equal to this set. The problem of the construction of functions with a given limit oscillations is formulated and investigated for the first time in the thesis. The limit oscillation of a locally constant function and real-valued continuous function, defined on an open subset of the metrizable space, are investigated. It is obtained the necessary and sufficient conditions of an arbitrary C-primitive (a function with a given cluster set) existence. The cluster sets of continuous functions defined on boundary locally connected open subsets of a metrizable space is characterized. It is characterized the cluster sets of quasi-locally constant functions.