# A function $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ that maps a circle to a circle

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?

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