If f and it's conjugate are analytic in a domain, then f is constant, Proof

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If f and it's conjugate are analytic in a domain, then f is constant, Proof. Please keep in mind that a lot of what should have been written down was said in words. You should always try to write your own proofs! I hope this helps someone out there.
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Why should be in connected domain? Whats happens in disconnected?

israeldeoliveirasilva
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An example showing that if f(z) is not const is merely continuous on a domain G, then f(G) need not be a domain?

soncknosferatum
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-v-       spooky. Good video though, easy to follow. :)

SirGumbi