  Let $T\subset\mathbb{R}$ and $(X,\mathcal{U})$ be a uniform space with an at
most countable gage of pseudometrics $\{d_p:p\in\mathcal{P}\}$ of the
uniformity $\mathcal{U}$. Given $f\in X^T$ (=the family of all functions from
$T$ into $X$), the approximate variation of $f$ is the two-parameter family
$\{V_{\varepsilon,p}(f):\varepsilon>0,p\in\mathcal{P}\}$, where
$V_{\varepsilon,p}(f)$ is the greatest lower bound of Jordan's variations
$V_p(g)$ on $T$ with respect to $d_p$ of all functions $g\in X^T$ such that
$d_p(f(t),g(t))\le\varepsilon$ for all $t\in T$. We establish the following
pointwise selection principle: If a pointwise relatively sequentially compact
sequence of functions $\{f_j\}_{j=1}^\infty\subset X^T$ is such that
$\limsup_{j\to\infty}V_{\varepsilon,p}(f_j)<\infty$ for all $\varepsilon>0$ and
$p\in\mathcal{P}$, then it contains a subsequence which converges pointwise on
$T$ to a bounded regulated function $f\in X^T$. We illustrate this result by
appropriate examples, and present a characterization of regulated functions
$f\in X^T$ in terms of the approximate variation.
