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== In mathematical analysis ==
The [[Banach fixed-point theorem]] (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of [[iteration|iterating]] a function yields a fixed point.<ref>{{cite book
| author = Giles, John R.
| title = Introduction to the Analysis of Metric Spaces
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}}</ref>
By contrast, the [[Brouwer fixed-point theorem]] (1911) is a non-[[Constructivism (mathematics)|constructive result]]: it says that any [[continuous function]] from the closed [[unit ball]] in ''n''-dimensional [[Euclidean space]] to itself must have a fixed point,<ref>Eberhard Zeidler, ''Applied Functional Analysis: main principles and their applications'', Springer, 1995.</ref> but it doesn't describe how to find the fixed point (See also [[Sperner's lemma]]).
For example, the [[cosine]] function is continuous in [−1,1] and maps it into [−1, 1], and thus must have a fixed point. This is clear when examining a sketched graph of the cosine function; the fixed point occurs where the cosine curve ''y'' = cos(''x'') intersects the line ''y'' = ''x''. Numerically, the fixed point is approximately ''x'' = 0.73908513321516 (thus ''x'' = cos(''x'') for this value of ''x'').
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