Cauchy-continuous function

Abstract: Some classes of metric spaces satisfying properties stronger than completeness but weaker than compactness have been studied by many authors over the years. One such significant family consists of those metric spaces on which every real-valued continuous function is uniformly continuous, which are widely known as Atsuji spaces or UC spaces. Recently in 2014, two new kinds of complete metric spaces are introduced, namely Bourbaki-complete and cofinally Bourbaki-complete metric spaces, whose idea has come from some new classes of sequences acting as generalizations of Cauchy sequences. Our major goal is to give several new equivalent conditions for metric spaces whose completions are one of the aforesaid spaces, especially in terms of some functions, sequences and geometric functionals.

Abstract: A real function is lacunary ideal ward continuous if it preserves lacunary ideal quasi Cauchy sequences where a sequence (x_{n}) is said to be lacunary ideal quasi Cauchy (or I_{θ}-quasi Cauchy) when (Δx_{n})=(x_{n+1}-x_{n}) is lacunary ideal convergent to 0. i.e. a sequence (x_{n}) of points in R is called lacunary ideal quasi Cauchy (or I_{θ}-quasi Cauchy) for every e>0 if {r∈N:(1/(h_{r}))∑_{n∈J_{r}}|x_{n+1}-x_{n}|≥e}∈I. Also we introduce the concept of lacunary ideal ward compactness and obtain results related to lacunary ideal ward continuity, lacunary ideal ward compactness, ward continuity, ward compactness, ordinary compactness, uniform continuity, ordinary continuity, δ-ward continuity, and slowly oscillating continuity. Finally we introduce the concept of ideal Cauchy continuous function in metric space and prove some results related to this notion.