Coffee without the Cup

Calculus Level 3

Suppose ff is a continuous function that maps the closed unit disk on R2\mathbb{R}^2 to itself. Then, Brouwer's fixed-point theorem tells us that there is a fixed point x0x_0 in the closed disk which is mapped to itself, i.e. f(x0)=x0f(x_0) = x_0.

In a similar spirit, let gg be a continuous function that maps the open unit disk to itself. Must gg have a fixed point?

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