Complex Analysis- Lecture 14 - Uniform continuity

Ex:-
If f(Z)=Z^2, then f is continuous at Z=i .show that?
If f(Z)=(3Z^4-2Z^3+8Z^4-2Z+5)/(Z-i), then f is discontinuous at Z=i.show that?
Is f(Z)=1/Z is continuous at Z=0 ? H.W
Is f(Z)=Z ?/Z is continuous at Z=0 ?
Sol:-

lim?(Z?Z_0 )??f(Z)=lim?(Z?i)??Z^2=(i)^2=-1? ?.
f(Z_0 )=Z_0^2=(i)^2=-1 must exist
lim?(Z?i) f(Z)=f(i)=-1
Thus f(Z)=Z^2 is continuous at Z_0=i .
f(i) doesn’t exist , i.e. f(Z) isn t defined at Z=i .Thus f(Z) isn t
Continuous at Z_0=i .
No,f(0) doesn’t exist , i.e. f(Z) isn t defined at Z=0 . Thus f(Z)
Isn t Continuous at Z_0=0 .
lim?(Z?Z_0 )??f(z)=lim?(Z?0)??Z ?/Z=lim??(x?0@y?0)??(x-iy)/(x+iy)? ? ?
lim??(x?0@y?0)??(x-iy)/(x+iy)=lim?(x?0)??(x-i(0))/(x+i(0))=? ? lim?(x?0) x/x=lim?(x?0) 1=1

lim??(x?0@y?0) (x-iy)/(x+iy)=lim?(y?0)??((0)-iy)/( (0)+iy)=lim?(y?0) (-iy)/iy=? lim?(y?0)-1=-1
We get the two limits are different. That is
lim?(Z?0)??Z ?/Z? isn t exist.

Continuity of region:- A function f(Z) is said to be continuous in a region if it is continuous at all points of the region .
Theorems on continuity :-
Theorem 1 :- If f(Z) and g(Z) are continuous at Z_0 , so also are the function f(Z)+g(Z) , f(Z)-g(Z), f(Z) g(Z) and f(Z)/g(Z) , the last only if g(Z_0 )?0 .
Similar results hold for continuity in a region.
Theorem 2 :- If f(Z) is continuous in a region, then the real and imaginary parts of f(Z) are also continuous in the region.
Uniform continuity:- Let f(Z) be continuous in a region. Then by definition at each point Z_0 of the region and for any ?>0 , we can find ?>0 such that |f(Z)-f(Z_0 ) |Alternatively, f(Z) is uniformly continuous in a region if for any ?>0, we can find ?>0 such that |f(Z_1 )-f(Z_2 ) |EX(1):- Prove that f(Z)=Z^2 is uniformly continuous in the region |Z|<1
Sol: - Let ?>0
|f(Z)-f(Z_0 ) |?|(Z+Z_0 )(Z-Z_0 ) |?|Z+Z_0 ||Z-Z_0 |?(|Z |+|Z_0 | )|Z-Z_0 |
?(|Z |+|Z_0 | )|Z-Z_0 |?2|Z-Z_0 |Where ? depends only on ? and not on? Z?_(0 ). Hence f(Z)=Z_0^2 is uniformly continuous in the region |Z|<1.
EX(2):- Prove that f(Z)=1/Z is not uniformly continuous in the region |Z|<1 .
Sol: - Suppose that f(Z) is uniformly continuous in the region.
Then for any ?>0 ,we should be able to find ? ,say between 0 and 1, such that |f(Z)-f(Z_0 ) |Let Z=? and Z_0=?/(1+?) , then |Z-Z_0 |=|?-?/(1+?)|=??/(1+?)However, |1/Z-1/Z_0 |=|1/?-(1+?)/?|=?/?>? (since 0Thus we have a contradiction, and it follow s that f(Z)=1/Z can t be uniformly continuous in the region.
H.W:- Prove that f(Z)=1/Z^2 is not uniformly continuous in the region |Z|?1.