= Solution
The complex <Liouville theorem> states that a bounded <entire function> is constant. Indeed, if $|f|\leq M$, the <Cauchy estimate> on any disc of radius $R$ centered at $z$ gives $|f'(z)|\leq M/R$. Letting $R\to\infty$ gives $f'(z)=0$ everywhere.
\b[The analogous conclusion for bounded <real analytic functions> on $\mathbb R$ is false.] For example, $\sin x$ is bounded, nonconstant, and <real analytic> on the whole real line. Boundedness only on that line does not bound its <holomorphic> extension on the complex plane.
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