Topological stability of continuous functions with respect to averaging by measures with locally constant densities

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.