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I can't find on complex analysys texts the precise definition of continuity at $\infty$. In particular, my lecturer said: all entire non costant functions aren't continous at infinity, since they are unbounded (by Liouville). I thought the following: take the complex plane with the point $\infty$, i.e. the Riemann sphere. This is a compact set. If f is entire non constant and i suppose $f$ to be continous also at $\infty$, then $f$ should be continous on $\mathbb C \cup\infty$, then bounded (by Weierstrass). Absurd. Is it right?

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    Maybe you didn't express yourself too well, but I understand your idea and I think that it is completely correct.2012-12-30

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Continuity at $\infty$ is usually understood to be continuity at the opposite of $0$ in Riemann sphere, that is correct (in general, study of meromorphic functions etc. makes sense in that context, in some ways more so than in the complex plane alone).

I don't understand what you mean by "absurd" -- you've simply shown that a complex-valued function holomorphic on $\bf C$ and continuous at $\infty$ is constant, and that is correct.

This may not be true if you allow the values to be $\infty$, too. Identity on the entire Riemann sphere is certainly holomorphic everywhere and nonconstant.

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    (Hmmm... I just saw all the typos in my first comment. Sorry about that.)2012-12-28