The 1-Dimensional Brouwer Fixed Point Theorem. It indicates that every continuous function mapping the closed interval
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The 1-Dimensional Brouwer Fixed Point Theorem. It indicates that every continuous function ƒ mapping the closed interval [0, 1] to itself must have a fixed point; that is, a point c such that (c) = c.
Show that if ƒ is continuous and 0 ≤ ƒ (x) ≤ 1 for 0 ≤ x ≤ 1, then ƒ(c) = c for some c in [0, 1] (Figure 7).
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If f0 0 the proof is done with c 0 We may assume that f0 0 Let gx fx x g0 f0 0 f0 0 Since f...View the full answer
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