Complex Analysis/Limits and continuity of complex functions

In this section, we introduce a 'broader class of limits' than known from real analysis (namely limits with respect to a subset of ) and characterise continuity of functions mapping from a subset of the complex numbers to the complex numbers using this 'class of limits'. Complex functions 0 100% #FFFFFF Definition 2.1: Let be sets and be a function. is a complex function if and only if . Example 2.2: The function is a complex function. Limits of complex functions with respect to subsets of the preimage We shall now define and deal with statements of the form for , , and , and prove two lemmas about these statements. 0 100% #FFFFFF Definition 2.3: Let be a set, let be a function, let , let and let . If , we define . 0 100% #FFFFFF Lemma 2.4: Let be a set, let be a function, let , let and . If , then . Proof: Let be arbitrary. Since , there exists a such that . But since , we also have , and thus , and therefore . 0 100% #FFFFFF Lemma 2.5: Let , be a function, be open, and . If , then for all such that . Proof: Let such that . First, since is open, we may choose such that . Let now be arbitrary. As , there exists a such that . We define and obtain . Continuity of complex functions 0 100% #FFFFFF Definition 2.6: Let and be a function. Then is defined to be continuous if and only if . Exercises <li>Prove that if we define , then is not continuous at . Hint: Consider the limit with respect to different lines through and use theorem 2.2.4. Next