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While studying analytic function i came across a problem which is:

If $D$ is the unit disc in $\mathbb C$ and $f:\mathbb C→D$ analytic with $f(10)=\frac{1}{2}$, then what is $f(10+i)$?

I think it is an application of Open mapping theorem which states that analytic functions which are non-constant map open sets to open sets."

My confusion is:By the only given condition $f(10)=\frac{1}{2}$, how can I deduce that $f$ is constant?. Is it not possible for a non- constant analytic function to attain a constant value at any point in its domain?

Any hint would be highly helpful. Thanks in advance.

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I think you need a different theorem to proof this:

https://en.wikipedia.org/wiki/Liouville%27s_theorem_%28complex_analysis%29

this gives you the answer directly

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  • $\begingroup$ @supinf..yes, here |f(z)|<1 implies f(z) is bounded and analytic hence it must be a constant function. Thanks a lot $\endgroup$ – Nitin Uniyal Jun 8 '15 at 5:09

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