Ministry of Higher Education
and Scientifics research
University of Babylon
College of Education for Pure Sciences
Physics Department
Three Stage
Complex Functions
Lecture 14
Ali Hussein Mahmood Al-Obaidi
ali.alobaidi81@yahoo.com
Ex:a. If
, then
is continuous at
b. If
, then
.show that?
is discontinuous at
.show
that?
c. Is
is continuous at
? H.W
d. Is
is continuous at
?
Sol:a.
1)
.
2)
3)
b.
must exist
Thus
is continuous at
.
doesn’t exist , i.e.
isn't defined at
Continuous at
c. No,
.Thus
.
doesn’t exist , i.e.
Isn't Continuous at
isn't defined at
.
d.
We get the two limits are different. That is
isn't exist.
1
. Thus
isn't
Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 14
College of Education for Pure Sciences
Physics Department
Three Stage
Ali Hussein Mahmood Al-Obaidi
ali.alobaidi81@yahoo.com
Continuity of region:- A function
is said to be continuous in a region
if it is continuous at all points of the region .
 Theorems on continuity :Theorem 1 :- If
function
and
,
are continuous at
,
and
, so also are the
, the last only if
.
Similar results hold for continuity in a region.
Theorem 2 :- If
is continuous in a region, then the real and imaginary
parts of
are also continuous in the region.
 Uniform continuity:- Let
be continuous in a region. Then by
definition at each point
of the region and for any
, we can find
such that
whenever
. If we can
find depending on but not the particular , we say that
is
uniformly continuous in the region.
Alternatively,
we can find
where
and
EX(1):- Prove that
is uniformly continuous in a region if for any
such that
whenever
are any two points of the region.
is uniformly continuous in the region
Sol: - Let
2
Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 14
College of Education for Pure Sciences
Physics Department
Three Stage
Ali Hussein Mahmood Al-Obaidi
ali.alobaidi81@yahoo.com
Where
Where depends only on
continuous in the region
and not on
.
EX(2):- Prove that
. Hence
is uniformly
is not uniformly continuous in the region
.
Sol: - Suppose that
Then for any
such that
region.
Let
and
is uniformly continuous in the region.
,we should be able to find
when
,say between
for all and
and ,
in the
, then
However,
Thus we have a contradiction, and it follow s that
can't be
uniformly continuous in the region.
H.W:- Prove that
is not uniformly continuous in the region
3