Let f: X → X be a function. A point x for which f(x) = x> is called a fixpoint of f.

If f is continuous, and if the sequence a, f(a), f(f(a)), f(f(f(a))), ... converges to some limit b, then b is a fixpoint of f. Unfortunately, the converse is not true: not every fixpoint is such a limit (only an attracting fixpoint). Additionally, the sequence a, f(a), f(f(a)), ... need not have a limit. For instance, if f is "rotate 32°" on R2, then for any a≠0 the sequence has no limit; despite this, a=0 is a fixpoint of f.

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