  Recently, a concept of forward continuity and a concept of forward
compactness are introduced in the senses that a function $f$ is forward
continuous if $\lim_{n\to\infty} \Delta f(x_{n})=0$ whenever $\lim_{n\to\infty}
\Delta x_{n}=0$,\; and a subset $E$ of $\textbf{R}$ is forward compact if any
sequence $\textbf{x}=(x_{n})$ of points in $E$ has a subsequence
$\textbf{z}=(z_{k})=(x_{n_{k}})$ of the sequence $\textbf{x}$ such that
$\lim_{k\to \infty} \Delta z_{k}=0$ where $\Delta z_{k}=z_{k+1}-z_{k}$. These
concepts suggest us to introduce a concept of second forward continuity in the
sense that a function $f$ is second forward continuous if
$\lim_{n\to\infty}\Delta^{2}f(x_{n})=0$ whenever
$\lim_{n\to\infty}\Delta^{2}x_{n}=0$, and a subset $E$ of $\textbf{R}$ is
second forward compact if whenever $\textbf{x}=(x_{n})$ is a sequence of points
in $E$ there is a subsequence $\textbf{z}=(z_{k})=(x_{n_{k}})$ of $\textbf{x}$
with $\lim_{k\to \infty} \Delta^{2}z_{k}=0$ where $\Delta^{2}
y_{n}=y_{n+2}-2y_{n+1}+y_{n}$. We investigate the impact of changing the
definition of convergence of sequences on the structure of forward continuity
in the sense of second forward continuity, and compactness of sets in the sense
of second forward compactness, and prove related theorems.
