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Corollary 9.5 (Brouwer's fixed point theorem):

Every continuous function $ f : E^2 \longrightarrow E^2$ has a fixed point where $ E^2 = \{ {\bf x} \in \mathbb{R}^2/ \Vert{\bf x} \Vert \leq 1 \}.$
nisha 2012-03-20