Letz0 C and assume f: C rightarrow C is continuous. If f is analytic on C righta

Question

Letz0 C and assume f: C rightarrow C is continuous. If f is analytic on C rightarrow {z0}, then probe f is entire. (Hint: Use Morera's Theorem. Need to look at f(z) dz where C is a triangle. Need to look at 3 cases for z0 Outside, on the boundary and inside)

Explanation / Answer

Given that f: C......> {z_0} is analytic. So it is constant function. Constant functions on C always entire functions. Infact you have given that f is analytic on C. Note: f is analytic on C means, f is entire. PLease let me know your question exactly. May be I am not getting. Try to explain me completely. Note: Due to f is analytic, Moreras theorem becomes trivial in this case.

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