Obstructions to the extension of partial maps

One of the most important problems in topology is the minimization (in some sense) of obstructions to extending a partial map $Z\hookleftarrow A\overset{f}{\to} X$, i.e., of a subset $F\subset Z\setminus A$ such that $f$ can be globally extended to its complement. It is shown that if $Z$ is a fixed metric space with $\dim Z\le n$ and $p,q\ge-1$ are fixed numbers, then obstructions to extending all partial maps $Z\hookleftarrow A\overset{f}{\to} X\in\operatorname{LC}^p\cap \operatorname{C}^q$ can be concentrated in preselected fairly thin subsets of $Z$.