Coffee without the Cup

Calculus Level 3

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

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

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