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Is there a function holomorphic on the open unit disk and continuoes on the closed disk such that f(z)= 1/z on the unit circle?

I will also like to know if somebody can help:

There are several approximation theorems out there, say Mergelyan, Runge, etc. Can somebody point at the salient features of these(i.e. when, what applies), or direct me to a source that is clear to a beginner. This sounds like spoon feeding, but I had to do it, bear with me. Thanks

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# Is there such a function?

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