Fixpoint

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function. That is to say, c is a fixed point of the function f(x) if and only if f(c) = c. This means f(f(...f(c)...)) = fⁿ(c) = c, an important terminating consideration when recursively computing f. A set of fixed points is sometimes called a fixed set.

For example, if f is defined on the real numbers by

then 2 is a fixed point of f, because f(2) = 2.

Not all functions have fixed points: for example, if f is a function defined on the real numbers as f(x) = x + 1, then it has no fixed points, since x is never equal to x + 1 for any real number. In graphical terms, a fixed point means the point (x, f(x)) is on the line y = x, or in other words the graph of f has a point in common with that line.

Points that come back to the same value after a finite number of iterations of the function are called periodic points. A fixed point is a periodic point with period equal to one. In projective geometry, a fixed point of a projectivity has been called a double point.

In Galois theory, the set of the fixed points of a set of field automorphisms is a field called the fixed field of the set of automorphisms.

An attractive fixed point of a function f is a fixed point x₀ of f such that for any value of x in the domain that is close enough to x₀, the iterated function sequence

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