$r_{\!\vphantom{\hat{A}}h}>0$ and $f>0$ for all $r>r_{\!\vphantom{\hat{A}}h}$).

For a given nonzero scalar field $\phi(r)$, it follows directly from (\[F-xi\]) that $\xi'=\mathrm{e}^{\Phi}>0$ for all $r>0$ and $\xi(0)>0$, so that the metric function $A$, given by the quadrature (\[A-f\]), passes through zero and becomes negative as $\,r\rightarrow0\,$ if and only if $\,\,3m>\xi(0)$. In other words, the corresponding configuration of mass $m$ will be a naked singularity or a black hole if $$\label{3m<xi0}
0<3m<\xi(0)\quad \mbox{(naked singularities)}$$ or $$\label{3m>xi0}
3m>\xi(0)\quad \mbox{(black holes)}$$ respectively. In what follows we deal only with ’generic’ configurations and do not consider the special (fine-tuned) case $3m=\xi(0)$; the latter leads to a naked singularity or a regular solution.

It follows from (\[F-xi\]) and (\[phi\]) that $$\begin{aligned}
\label{est1}
\mathrm{e}^{\Phi}&=&1+o(1/r),\quad \xi\!=r+o(1),\;\;\; r\rightarrow\infty,\vphantom{\int}\\
\label{est2}
\mathrm{e}^{\Phi}&=&\mathrm{e}^{\Phi(0)}+o(r),\;\; \xi\!=\xi(0)+\xi'(0)r+o(r),\;\; r\rightarrow0,\end{aligned}$$ where $\xi(0)>0$ and $\xi'(0)=\mathrm{e}^{\Phi(0)}>0$ if the scalar field is not identically zero. Using the quadrature (\[A-f\]) and the asymptotic estimates (\[est1\]) and (\[est2\]), one can directly obtain the asymptotic behaviours of the metric function $A(r)$ for both SFNSs and SFBHs; they are $$\begin{aligned}
\label{asymp1}
A(r)&=&1-\frac{2m}{r}\vphantom{\int\limits_a}+ o(1/r),\;\; r\rightarrow\infty,\\
\label{asymp2}
A(r)&=&\frac{2}{3}\:\! \frac{\xi(0)-3m}{r}\,\mathrm{e}^{\Phi(0)}\,+\, O(1),\;\; r\rightarrow0.\end{aligned}$$ The condition $m>0$ distinguishes SFNSs from the vacuum (that is, Schwarzschild) naked singularities, which exist only for negative values of mass. Below we will also 