Strongly Cesaro type quasi-Cauchy sequences

In this paper we call a real-valued function $N_{\theta}$-ward continuous if it preserves $N_{\theta}$-quasi-Cauchy sequences where a sequence $\boldsymbol{\alpha}=(\alpha_{k})$ is defined to be $N_{\theta}$-quasi-Cauchy when the sequence $\Delta \boldsymbol{\alpha}$ is in $N^{0}_{\theta}$. We prove not only inclusion and compactness type theorems, but also continuity type theorems.