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 ? .
example:
suppose f(z)=x^2-y^2+ixysin(x ) ?
then f is continuous on ?
suppose f(z)=e^xy-e^(y^2/2)+i sin?(x^2-2y+3)
then f is continuous on ?
lemma?-
suppose f,g are contiuous function on R then f+g ,f-g ,f.g ,f/g (g?0)
are all continuous functions on R.
lemma?-
composite two continuous function is again a continuous function .
example:
e^2z is continuous on ?.
Note?-
suppose f is continuos on C
|f(z)| ,f(z ? ),(f(z) ) ? are continuos function on C .
example
suppose f(z)=2z^2+iz then f is continuous on ? ,
because it is polynomial
example:
suppose f(z)=z/(z^2+4),it is continuous on ? ,except z=±2i
example:
f(z)=z ?/z,it is not continuous when z=0
since lim?(z?0)??z ?/z? is not exist .
DERIVATIVIES:-
let f be a function defined on adomain containing anbhd of apoint
z_0 ,then the derivative of f at z_0 ,denoted by f ?(z_0 ) is defined by
f ?(z_0 )=lim?(z?z_0 )??(f(z)-f(z_0 ))/(z-z_0 )?
provided the limit exists .
f is said to be differentiable at z_0 if the derivative f ?(z_0 ) exists .
set ,?z=z-z_0
then f ?(z_0 )=lim?(?z?0)??(f(z_0+?z)-f(z_0 ))/?z?
equavalently,
f ?(z) =lim?(?z?0)??(f(z+?z)-f(z))/?z?
here |?z| is so small that f(z_0+?z)is defined