is $\rho$-continuous if and only if it is continuous with respect to $H(\rho)$ as a usual map, namely if for every $\varepsilon>0$, each point $x\in X$ is contained in an open set $V$ such that $H(\rho)(\varphi(p),\varphi(x))<\varepsilon$ for every $p\in V$.

\[proposition-Finite-C-v4:1\] Let $\varphi:X\operatorname{\leadsto}Y$ be a $\rho$-continuous mapping on a space $X$, and $x\in X$ be such that $\varphi(x)$ is uniformly $UV^*$. Then for every $\varepsilon>0$ there exists $\delta_x\in (0,\varepsilon)$ and an open set $U_x\subset X$ containing $x$ such that for every $p,q\in U_x$, $$H(\rho)(\varphi(p),\varphi(q))<\frac{\delta_x}2\quad
    \text{and}\quad \mathbf{O}_{\delta_x}(\varphi(p)){\lhook\joinrel\xrightarrow{*}}
    \mathbf{O}_\varepsilon(\varphi(p)).$$

Take $\delta_x\in \left(0,\frac\varepsilon4\right)$ with $\mathbf{O}_{2\delta_x}(\varphi(x)) {\lhook\joinrel\xrightarrow{*}}
  \mathbf{O}_{\frac\varepsilon2}(\varphi(x))$, and set $$U_x=\left\{p\in X:
      H(\rho)(\varphi(p),\varphi(x))<\frac{\delta_x}4\right\}.$$ Then $U_x$ is an open set such that $H(\rho)(\varphi(p),\varphi(q))<\frac{\delta_x}2$, for every $p,q\in U_x$. Moreover, $p\in U_x$ implies $\mathbf{O}_{\delta_x}(\varphi(p)){\lhook\joinrel\xrightarrow{*}}
  \mathbf{O}_\varepsilon(\varphi(p))$ because ${\lhook\joinrel\xrightarrow{*}}$ is monotone and $$\mathbf{O}_{\delta_x}(\varphi(p))\subset
    \mathbf{O}_{2\delta_x}(\varphi(x)) {\lhook\joinrel\xrightarrow{*}}
    \mathbf{O}_{\frac\varepsilon2}(\varphi(x)) \subset
    \mathbf{O}_\varepsilon(\varphi(p)).\qedhere$$

\[proposition-Finite-C-v4:3\] Let $X$ be a paracompact space and $\varphi:X\operatorname{\leadsto}Y$ be a $\rho$-continuous mapping such that each $\varphi(x)$, $x\in X$, is uniformly $UV^*$. Suppose that $\mathscr{U}_1$ is a locally finite open cover of $X$ and $\delta_1:\mathscr{U}_1\to (0,+\infty)$. Then there exists an open locally finite refinement $\mathscr{U}_2$ of $\mathscr{U}_1$ and functions $\delta_2,\varepsilon_2:\mathscr{U}_2\to (0,+\infty)$ with $\delta_2\leq \varepsilon_2$, such that

1.  $\mathbf{O}_{\delta_2(U)}(\varphi(p)){\lhook\joinrel\xrightarrow{*}}
        \mathbf{O}_{\varepsilon_2(U)}(\varphi(p))$, for every $p\in U\in \mathscr{U}_2$,

2.  $H(\rho)(\varphi(p),\varphi(q))<\frac{\delta_2(U)}2$, for every $p,q\in U\in \mathscr{U}_2$,

