On the unit disk f(z)=Σₙ₌₀∞a_nzⁿ, where a₀=a₁=1 and a_n=1/2^(2n) for n≥2. P:f is one-to-one in some neighborhood of every disk point. Q:f({|z|≤1/2}) is closed in ℂ. Select all correct statements.
For |z|<1, $|f\prime(z)-1|\le\sum_{n\ge2}n/4^n=4/9-1/4=7/36<1$. Thus f′ never vanishes, and the analytic inverse-function theorem gives local injectivity. The half-radius closed disk is compact; its continuous image is compact and hence closed in ℂ. **A,B**.
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