Hölder continuity of $p(x)$-harmonic functions

The question on the Hölder continuity of solutions of the $p$-Laplace equation with measurable summability index $p=p(x)$ bounded away from one and infinity is studied. In the case when the domain of definition $D\subset\mathbb R$, $n\geqslant2$, of the equation is partitioned by a hyperplane $\Sigma$ into parts $D^{(1)}$ and $D^{(2)}$ such that $p(x)$ has a logarithmic modulus of continuity at a point $x_0\in D\cap\Sigma$ from either side it is proved that solutions of the equation are Hölder-continuous at $x_0$. The case when $p(x)$ has a logarithmic modulus of continuity in $D^{(1)}$ and $D^{(2)}$ is considered separately. It is proved that smooth functions in $D$ are dense in the class of solutions.