A new variation on statistical ward continuity

A real valued function defined on a subset $E$ of $\mathbb{R}$, the set of real numbers, is $\rho$-statistically downward continuous if it preserves $\rho$-statistical downward quasi-Cauchy sequences of points in $E$, where a sequence $({\alpha }_k)$ of real numbers is called $\rho$-statistically downward quasi-Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{\rho_{n} }|{k\leq n: \Delta \alpha_{k} \geq \varepsilon}|=0 $forevery$\varepsilon>0$, in which $({\rho }_n)$ is a non-decreasing sequence of positive real numbers tending to $\mathrm{\infty }$ such that ${\mathrm{lim\hspace{0.17em}sup}}_n\frac{{\rho }_n}{n}<\mathrm{\infty }$, $\mathrm{\Delta }{\rho }_n=O(1)$, and $\mathrm{\Delta }{\alpha }_k={\alpha }_{k+1}-{\alpha }_k$ for each positive integer $k$. It turns out that a function is uniformly continuous if it is $\rho$-statistical downward continuous on an above bounded set.