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For f : $\mathbb R^2$ → $\mathbb R$ a continuous function and D the closed unit disc in $\mathbb R^2$. Is f(D) necessarily an interval in $\mathbb R$? If it is an interval, which of the forms $]a, b[, [a, b[, ]a, b]$ and $[a, b]$, with $a, b \in\mathbb R$ can it have?

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up vote 3 down vote accepted

HINT: Continuous functions take compact sets to compact sets. A set in $\Bbb R^n$ is compact if and only if it is closed and bounded.

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Also, continuous functions take connected sets to connected sets. – David Mitra Jan 18 '13 at 16:59

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