The Correction Theorem for Anisotropic Spaces

The following old problem is solved. Given an $\varepsilon>0$, a function $f \colon [0,1]^n\to\mathbb R$, and the partial moduli of continuity of this function evaluated in a symmetric space $X$, find a set $I(\varepsilon)$ of measure larger than $1-\varepsilon$ such that the partial uniform moduli of continuity of f determined for the points of this set admit an unimprovable (with respect to all restrictions to sets of measure larger than $1-\varepsilon$) estimate of partial uniform moduli of continuity and write out this estimate of the uniform partial moduli of continuity.