scholarly journals Distance to Spaces of Semicontinuous and Continuous Functions

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Carlos Angosto
Keyword(s):  
Topological Space ◽  
Lower Semicontinuous ◽  
Continuous Functions ◽  
Upper Semicontinuous ◽  
Lower Semicontinuous Functions ◽  
Upper Semicontinuous Functions ◽  
Semicontinuous Functions

Given a topological spaceX, we establish formulas to compute the distance from a functionf∈RXto the spaces of upper semicontinuous functions and lower semicontinuous functions. For this, we introduce an index of upper semioscillation and lower semioscillation. We also establish new formulas about distances to some subspaces of continuous functions that generalize some classical results.

Real Functions, Covers and Bornologies

2021 ◽  
Vol 78 (1) ◽  
pp. 199-214
Author(s):  
Lev Bukovský
Keyword(s):  
Topological Space ◽  
Pointwise Convergence ◽  
Continuous Functions ◽  
Upper Semicontinuous ◽  
Upper Semicontinuous Functions ◽  
Covering Properties ◽  
Real Functions ◽  
Topology Of Pointwise Convergence ◽  
Semicontinuous Functions

Abstract The paper tries to survey the recent results about relationships between covering properties of a topological space X and the space USC(X) of upper semicontinuous functions on X with the topology of pointwise convergence. Dealing with properties of continuous functions C(X), we need shrinkable covers. The results are extended for A-measurable and upper A-semimeasurable functions where A is a family of subsets of X. Similar results for covers respecting a bornology and spaces USC(X) or C(X) endowed by a topology defined by using the bornology are presented. Some of them seem to be new.

A unified approach to continuous and certain non-continuous functions

1990 ◽  
Vol 48 (3) ◽  
pp. 347-358 ◽  
Author(s):  
J. K. Kohli
Keyword(s):  
Lower Semicontinuous ◽  
Continuous Functions ◽  
Unified Theory ◽  
Unified Approach ◽  
Lower Semicontinuous Functions ◽  
Almost Continuous ◽  
Semicontinuous Functions

AbstractA unified theory of continuous and certain non-continuous functions is proposed and developed. The proposed theory encompasses in one the theories of continuous functions, upper (lower) semicontinuous functions, almost continuous functions, c-continuous functions, c*-continuous functions, s-continuous functions, l-continuous functions, H-continuous functions, and the ε-continuous functions of Klee.

scholarly journals A unified approach to continuous and certain non-continuous functions II

1990 ◽  
Vol 41 (1) ◽  
pp. 57-74 ◽  
Author(s):  
J.K. Kohli
Keyword(s):  
Functional Analysis ◽  
Continuous Functions ◽  
Unified Theory ◽  
Unified Approach ◽  
Lower Upper ◽  
Upper Semicontinuous ◽  
Weakly Continuous ◽  
Upper Semicontinuous Functions ◽  
Almost Continuous ◽  
Semicontinuous Functions

A unified theory of continuous and certain non-continuous functions, initiated in an earlier paper, is further elaborated. The proposed theory provides a common platform for dealing simultaneously with continuous functions and a host of non-continuous functions including lower (upper) semicontinuous functions, almost continuous functions, weakly continuous functions (encountered in functional analysis), c-continuous functions, δ-continuous functions, semiconnected functions, H-continuous functions s-continuous functions, ε-continuous functions of Klee and several other variants of continuity.

scholarly journals Approximations by differences of lower semicontinuous functions

2015 ◽  
Vol 62 (1) ◽  
pp. 183-190 ◽  
Author(s):  
Eduard Omasta
Keyword(s):  
Topological Space ◽  
Lower Semicontinuous ◽  
Classical Theorem ◽  
Abstract Theorem ◽  
Lower Semicontinuous Functions ◽  
Baire One ◽  
Semicontinuous Functions

Abstract A classical theorem of W. Sierpiński, S. Mazurkiewicz and S. Kempisty says that the class of all differences of lower semicontinuous functions is uniformly dense in the space of all Baire-one functions. We show a generalization of this result to more general situations and derive an abstract theorem in the case of a binormal topological space.

Sandwich Theorems for Semicontinuous Operators

1992 ◽  
Vol 35 (4) ◽  
pp. 463-474 ◽  
Author(s):  
J. M. Borwein ◽  
M. Théra
Keyword(s):  
Lower Semicontinuous ◽  
Lower Semicontinuous Functions ◽  
Classical Interpolation ◽  
Semicontinuous Functions

AbstractWe provide vector analogues of the classical interpolation theorems for lower semicontinuous functions due to Dowker and to Hahn and Katetov-Tong.

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