Brouwer fixed-point theorem: Difference between revisions

Revision as of 13:31, 14 June 2025

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Brouwer "flattens" his sheet as with a flat iron, without removing the folds and wrinkles. Unlike the coffee cup example, the crumpled paper example also demonstrates that more than one fixed point may exist. This distinguishes Brouwer's result from other fixed-point theorems, such as [[Stefan Banach]]'s, that guarantee uniqueness.

===TwoOne-dimensional case===

[[File:Théorème-de-Brouwer-dim-1.svg|200px|right]]

In theone two dimensional planedimension, the result is intuitive and easy to prove. The continuous function ''f'' is defined on a closed interval [''a'', ''b''] and takes values in the same interval. Saying that this function has a fixed point amounts to saying that its graph (dark green in the figure on the right) intersects that of the function defined on the same interval [''a'', ''b''] which maps ''x'' to ''x'' (light green).

Intuitively, any continuous line from the left edge of the square to the right edge must necessarily intersect the green diagonal. To prove this, consider the function ''g'' which maps ''x'' to ''f''(''x'') − ''x''. It is ≥ 0 on ''a'' and ≤ 0 on ''b''. By the [[intermediate value theorem]], ''g'' has a [[Root of a function|zero]] in [''a'', ''b'']; this zero is a fixed point.