# Prove that $f$ is a diffeomorphism $C^∞$, that maps... (More inside)

Prove that $f:(x, y) ? R^2$ - {(0, 0)} ? ($\frac{x}{x^2 +y^2}$ ; $\frac{y}{x^2+y^2}$) ? $R^2$ - {(0, 0)} is a diffeomorphism $C^?$, that maps each circle of radius $r > 0$ centered in the origin to a cocentric circle with radius $\frac{1}{r}$ .

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