concept of limits in complex plane

Chapter two concept of limit point
suppose f is defined at all points in some nbhd of a point z_0 ,by the statment
that
lim?(z?z_0 )??f(z)=w_0 …(*)?
we mean that w=f(z) can be made arbitrary close to w_0 provided z
is sufficiently close to z_0 and distinct from it.
mathematically:-
this means for each ?>0 arbitarly small there exist ?>0 ?
|f(z)-w_0 |i.e. ???(?,zo).
Geometrically:-
this means for each ?-nbhd |w-w_0 |?-nbhd of z_0 such that each point of this a deleted ?-nbhd of z_0 such that each point of this nbhd has an image lying in the ?-nbhd of w_0 since (*)holds for each point of the nbhd we have
z goes to z_0 in any path .
f should be define at all point
there must be the connectedness
notice that nbhd of z_0 comes into enistance if z_0 is an interior point .
(in particular in the case of an open set)
by the uniform limit we mean ? depends only on ?.(i.e independent of z_0 )