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Constant functions are holomorphic without preserving the properties of C

The issue is that the Cauchy Riemann equations don't hold. (You would need 1==-1)

Picard's theorem tells us constant functions are the only example of that, though (more or less because 0 == -0 [1] and that's only true for 0).

[1] I sincerely apologize to anyone who works with floats for this statement.

No need to apologize, 0 == -0 even if we are talking about floats.