  We prove continuity and Harnack's inequality for bounded solutions to the
equation $$ {\rm div}\big(|\nabla u|^{p(x)-2}\,\nabla u \big)=0, \quad p(x)= p
+ L\frac{\log\log\frac{1}{|x-x_{0}|}}{\log\frac{1}{|x-x_{0}|}},\quad L > 0, $$
under the precise non-logarithmic condition on the function $p(x)$.
