continuity

continuity:
definition: let ? be defined on a nbhd of z_0 then f is said to be continuous at z_0
if lim?(z?z_0 )??f(z)=f(z_0 )?
notice that in it ,we have
lim?(z?z_0 )?f(z) exist
f(z_0 ) exist
lim?(z?z_0 )??f(z)=f(z_0 )?
so f is continnous at z_0 if for each ?>0 ? ??? (?,z_0)>0
such that |z-z_0 |if f is continuous at each point of a region ? then we have say that f is continuous
on ? .
?definition:??? uniformly continuous?-?
f is said to be uniformly continuous on ? .
if for each ?>0 ? ?>0 ,? ??(?)
|f(z_1 )-f(z_2)|for each pair of points z_1,z_2 in ?.
note that uniformly continious on R leads to continuity on R
but converse is not true .
and we proof composite two continuous function is again a continuous function,suppose f is continuous at z_0 then there exists anbhd of z_0 on which f is
,Let R be a closed and bounded region in ? ,suppose f is continuous on R .
then
f is bounded on R.
f is uniformly continuous on R

non-zero provided f(z)?0.