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A function $f$ defined on a subset $E$ of a two normed space $X$ is $S_{\theta}$-ward (respectively, $N_{\theta}$-ward) continuous if it preserves $S_{\theta}$-quasi-Cauchy (respectively, $N_{\theta}$-quasi-Cauchy) sequences of points in $E$; that is, a sequence $(f(x_k))$ is an $S_{\theta}$-quasi-Cauchy (respectively, an $N_{\theta}$-quasi-Cauchy) sequence whenever $(x_{k})$ is $S_{\theta}$-quasi-Cauchy (respectively, $N_{\theta}$-quasi-Cauchy). A subset $E$ of $X$ is $S_{\theta}$-ward (respectively, $N_{\theta}$-ward) compact if any sequence of points in $E$ has an an $S_{\theta}$-quasi-Cauchy (respectively, $N_{\theta}$-quasi-Cauchy) subsequence. In this paper, not only $S_{\theta}$-ward (respectively, $N_{\theta}$-ward) continuity, but also some other kinds of continuities are investigated in two normed spaces. It turns out that uniform limit of $S_{\theta}$-ward (respectively, $N_{\theta}$-ward) continuous functions is again $S_{\theta}$-ward (respectively, $N_{\theta}$-ward) continuous.

Lacunary statistical convergence; $N_{\theta}$ convergence; quasi-Cauchy sequences; continuity