Complex Analysis/Complex Functions/Continuous 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'. Limits of complex functions with respect to subsets of the preimage We shall now define and deal with statements of the form for , and prove two lemmas about these statements. 0 100% #FFFFFF Definition 2.2.1: Let be a set, let be a function, let , let and let . If we define: 0 100% #FFFFFF Lemma 2.2.2: 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 2.2.3: 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 We recall that a function where are metric spaces, is continuous if and only if for all convergent sequences in . 0 100% #FFFFFF Theorem 2.2.4: Let and be a function. Then is continuous if and only if Proof Exercises 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