Suppose f is a continuous function from $\mathbb{R}^2$ to $\mathbb{R}^2$ that maps a circle to a circle. How do I prove that f is differentiable? Will the function still be differentiable if continuity is not assumed?

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    $\begingroup$ Any circle to any circle, or one particular circle? $\endgroup$ – Randall Oct 11 '18 at 15:01
  • $\begingroup$ @Randall Any circle. $\endgroup$ – Prabhat Oct 11 '18 at 15:05
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    $\begingroup$ Interesting. Does it map any circle to a circle or any circle to itself? Big difference. $\endgroup$ – Randall Oct 11 '18 at 15:07
  • $\begingroup$ Any circle to a circle. $\endgroup$ – Prabhat Oct 11 '18 at 15:12
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    $\begingroup$ How did you come across this problem? $\endgroup$ – Calvin Khor Oct 11 '18 at 17:19

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