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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