On a nonempty connected open Ω⊂ℂ, f is nonconstant. S1: if f is continuous and f² analytic, f is analytic. S2: if f² and f³ are analytic, f is analytic. Choose truth values.
For S1, away from zeros of f, a local analytic square root of nonzero f² exists and continuity selects a fixed sign, making f analytic locally. Zeros of the nonconstant analytic f² are isolated; f is continuous/bounded there, so removable singularity applies. For S2, off those zeros f=f³/f² is analytic, and |f|²=|f²| bounds f near each zero, again making it removable. **Both true, C**.
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