Let X and Y be metric spaces.
The goal is to study some intermediate classes of functions
between the class $C$_{uc}(X,Y)
of all uniformly
continuous mappings (briefly, UC) from X into Y
and the class C(X,Y) of all continuous functions f:X->Y.
These classes, defined below, have been intensively studied
in earlier papers mainly in the case when Y=**R**, the real line.
In this paper we study them for general Y. In particular,
we will consider the case when X=Y is the complex plane C.

- For subsets K,M of X we say that g:X->Y is a (K,M)-approximation of f if g is a UC map such that g[M] is a subset of f[M] and g(x)=f(x) for each x from K.
- Function f is uniformly approachable (briefly, UA) if f has a (K,M)-approximation for every compact subset K of X and every subset M of X.
- Function f is weakly uniformly approachable
(briefly, WUA) if f has a (x,M)-approximation
for every x form X and every subset M of X.

(*)
$C$_{uc}(X,Y)-->C_{ua}(X,Y)-->
C_{wua}(X,Y)-->C(X,Y),

where the arrow --> stands for the inclusion. Between other things we discuss possible equations in (*) for different spaces X and Y.

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**Last modified October 20, 2001.**