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 $\mathit{\alpha }=({\alpha }_k)$ is defined to be $N_{\theta }$-quasi-Cauchy when the sequence $\mathrm{\Delta }\mathit{\alpha }$ is in $N_{\theta }^0$. We prove not only inclusion and compactness type theorems, but also continuity type theorems.