This paper proves that for every Lipschitz function
$f:{\bb R}^n\longrightarrow {\bb R}^m,\;m < n$
, there exists at least one point of
$\varepsilon$
-differentiability of
$f$
which is in the union of all
$m$
-dimensional affine subspaces of the form
$q_0+{\rm span}\{q_1,q_2,\ldots,q_m\},\;{\rm where}\;q_j(j=0,1,\ldots,m)$
are points in
${\bb R}^n$
with rational coordinates.