  We introduce and study adhesive spaces. Using this concept we obtain a
characterization of stable Baire maps $f:X\to Y$ of the class $\alpha$ for wide
classes of topological spaces. In particular, we prove that for a topological
space $X$ and a contractible space $Y$ a map $f:X\to Y$ belongs to the $n$'th
stable Baire class if and only if there exist a sequence $(f_k)_{k=1}^\infty$
of continuous maps $f_k:X\to Y$ and a sequence $(F_k)_{k=1}^\infty$ of
functionally ambiguous sets of the $n$'th class in $X$ such that
$f|_{F_k}=f_k|_{F_k}$ for every $k$. Moreover, we show that every monotone
function $f:\mathbb R\to \mathbb R$ is of the $\alpha$'th stable Baire class if
and only if it belongs to the first stable Baire class.
