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In numerical analysis, fixed-point iteration is a method of computing fixed points of iterated functions.
More specifically, given a function f defined on the real numbers with real values and given a point x_0 in the domain of f, the fixed point iteration is
which gives rise to the sequence x_0, x_1, x_2, \dots which is hoped to converge to a point x. If f is continuous, then one can prove that the obtained x is a fixed point of f, i.e.,
More generally, the function f can be defined on any metric space with values in that same space.
converges to 0 for all values of x_0. However, 0 is not a fixed point of the function
as this function is not continuous at x=0, and in fact has no fixed points.
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