Here a question that has me stumped. Maybe someone familiar with algebraic or differential curves can help. Suppose that $\gamma:[0,1] \rightarrow \mathbb{C}$ is an analytic function. Is it true that the range of $\gamma$ is either homeomorphic to a line segment or contains a subset homeomorphic to $S^1$?
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@Richard: Real analytic is the same as complex analytic since locally the power series expansions converge on disks. Yes, $\gamma$ is analytic (or has analytic extension) to a neighborhood of $[0,1]$. 

