# Type of isolated singularity of holomorphic funtion unique?

I am using "Real and complex analysis" by rudin. I have just read the proof concerning the three different types of isolated singularities, and that one of them must occur. The book doesn't say anything about these three types being mutually exclusive. Is it trivial? It doesn't seem so obvious to me. thanks

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Either $\lim_{z\to z_0} f(z)$ exists and is finite (removable), goes to infinity in norm (pole), or it doesn't exist (essential). What else could happen to the limit? –  Jose27 Dec 2 '12 at 20:24