A study on \(N_\theta\)-quasi-Cauchy sequences (Q370255)

Summary: Recently, the concept of \(N_\theta\)-ward continuity was introduced and studied. In this paper, we prove that the uniform limit of \(N_\theta\)-ward continuous functions is \(N_\theta\)-ward continuous, and the set of all \(N_\theta\)-ward continuous functions is a closed subset of the set of all continuous functions. We also obtain that a real function \(f\) defined on an interval \(E\) is uniformly continuous if and only if \((f(\alpha_k))\) is \(N_\theta\)-quasi-Cauchy whenever \((\alpha_k)\) is a quasi-Cauchy sequence of points in \(E\).