Can anyone point me out the continuous functions without brouwer fixed point's for the following sets
$$A = \{x \in \mathbb{R}^2 | x_1,x_2 \geq 0 \text{ and }x_1^2+x_2^2 = 1 \}$$ $$B = \{x \in \mathbb{R}^2 | x_1,x_2 \geq 0 \text{ and }2 \geq x_1^2+x_2^2 \geq 1 \}$$ $$C = \{x \in \mathbb{R}^2 | x_1^2+x_2^2 < 1 \}$$
How should one be designing continuous functions without fixed point? Is there any specific method is it just a guess?
$A$ is homeomorphic to the line segment $[-1,1]$, which always has fixed points.
$B$: rotate.
$C$ is homeomorphic to the plane. In the plane, a translation will do.