  A real valued function $f$ defined on a subset $E$ of $\textbf{R}$, the set
of real numbers, is statistically upward continuous if it preserves
statistically upward half quasi-Cauchy sequences, is statistically downward
continuous if it preserves statistically downward half quasi-Cauchy sequences;
and a subset $E$ of $\textbf{R}$, is statistically upward compact if any
sequence of points in $E$ has a statistically upward half quasi-Cauchy
subsequence, is statistically downward compact if any sequence of points in $E$
has a statistically downward half quasi-Cauchy subsequence where a sequence
$(x_{n})$ of points in $\textbf{R}$ is called statistically upward half
quasi-Cauchy if \[ \lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n:
x_{k}-x_{k+1}\geq \varepsilon\}|=0 \] is statistically downward half
quasi-Cauchy if \[ \lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n:
x_{k+1}-x_{k}\geq \varepsilon\}|=0 \] for every $\varepsilon>0$. We investigate
statistically upward continuity, statistically downward continuity,
statistically upward half compactness, statistically downward half compactness
and prove interesting theorems. It turns out that uniform limit of a sequence
of statistically upward continuous functions is statistically upward
continuous, and uniform limit of a sequence of statistically downward
continuous functions is statistically downward continuous.
