Quasi-Cauchy Sequences on θ-Parametric Metric Spaces

This paper proposes quasi-Cauchy sequences in θ-parametric metric spaces, a generalized metric framework introduced by Moussaoui et al. in 2024. It establishes fundamental relationships among convergent, Cauchy, and quasi-Cauchy sequences, and puts forward concepts of ward compactness and ward continuity in θ-parametric metric spaces. Moreover, the present paper demonstrates that ward continuity on a ward-compact subset in which every quasi-Cauchy sequence admits a convergent subsequence implies ordinary continuity, that uniformly continuous functions are necessarily ward continuous, and that uniform limits of ward-continuous functions preserve ward continuity. Consequently, all ward-continuous functions on E form a closed set of continuous functions on E.