[en] The question on the Hoelder 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 of R, n≥2, of the equation is partitioned by a hyperplane Σ into parts D⁽¹⁾ and D⁽²⁾ such that p(x) has a logarithmic modulus of continuity at a point x₀ element of D intersection Σ from either side it is proved that solutions of the equation are Hoelder-continuous at x₀. 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.