  Let $\mu$ be a measure on $[-1,1]$. Then for every continuous function
$f:\mathbb{R}\to\mathbb{R}$ and $\alpha>0$ one can define its averaging
$f_{\alpha}:\mathbb{R}\to\mathbb{R}$ by the formula: \[ f_{\alpha}(x) =
\int_{-1}^{1} f(x+t\alpha)d\mu. \] In arXiv:1509.06064 the authors studied the
problem when $f_{\alpha}$ is topologically equivalent to $f$ for all $\alpha>0$
and call this property a topological stability of $f$ under averagings with
respect to measure $\mu$. Similarly one can define topological stability of a
germ of $f$ at some point $x\in\mathbb{R}$.
  It was shown that for a continuous function $f:\mathbb{R}\to\mathbb{R}$
having only finitely many local extremes and any measure $\mu$ topological
stability of averagings of germs of $f$ at local extremes implies topological
stability of averagings of $f$. In the present paper we prove the converse
statement: topological stability of averagings of $f$ implies topological
stability of averagings of its germs at local extremes.
  Moreover, in arXiv:1509.06064 it was also extablished a sufficient condition
for topological stability of averagings of local extremes with respect to
measures with finite supports. In the present paper we obtain similar
sufficient conditions for measures with locally continuous and in particular
with locally constant densities.
