Discussion
Let \(f\) be a non-constant entire function. Then which of the following is impossible ?
Answer: D
The impossible statement is \(f(\mathbf{C}) \subseteq \mathbf{R}\).
By the open mapping theorem, a non-constant entire function maps C onto an open set, and no non-empty open subset of the plane lies inside the real line. (Alternatively, if \(f\) were real, \(e^{if}\) would be a bounded entire function, hence constant by Liouville.)
The others occur: \(f(z) = z\) is onto C (A); \(e^{z}\) omits exactly 0 (B), consistent with Picard's theorem; and the image of any non-constant entire function is dense in C (C), by the Casorati-Weierstrass argument.
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