  Let $T\subset\mathbb{R}$, $M$ be a metric space with metric $d$, and $M^T$ be
the set of all functions mapping $T$ into $M$. Given $f\in M^T$, we study the
properties of the approximate variation $\{V_\varepsilon(f)\}_{\varepsilon>0}$,
where $V_\varepsilon(f)$ is the greatest lower bound of Jordan variations
$V(g)$ of functions $g\in M^T$ such that $d(f(t),g(t))\le\varepsilon$ for all
$t\in T$. The notion of $\varepsilon$-variation $V_\varepsilon(f)$ was
introduced by Fra\'nkov\'a [Math. Bohem. 116 (1991), 20-59] for intervals
$T=[a,b]$ in $\mathbb{R}$ and $M=\mathbb{R}^N$ and extended to the general case
by Chistyakov and Chistyakova [Studia Math. 238 (2017), 37-57]. We prove
directly the following basic pointwise selection principle: If a sequence of
functions $\{f_j\}_{j=1}^\infty$ from $M^T$ is such that the closure in $M$ of
the set $\{f_j(t):j\in\mathbb{N}\}$ is compact for all $t\in T$ and
$\limsup_{j\to\infty}V_\varepsilon(f_j)$ is finite for all $\varepsilon>0$,
then it contains a subsequence, which converges pointwise on $T$ to a bounded
regulated function $f\in M^T$. We establish several variants of this result for
sequences of regulated and nonregulated functions, for functions with values in
reflexive separable Banach spaces, for the almost everywhere convergence and
weak pointwise convergence of extracted subsequences, and comment on the
necessity of assumptions in the selection principles. The sharpness of all
assertions is illustrated by examples.
