Fixed Point (mathematics)

Fixed Point (mathematics)

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set. That is to say, c is a fixed point of the function f(x) if and only if f(c) = c. 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. The example f(x) = x + 1 is a case where the graph and the line are a pair of parallel lines.

Points which come back to the same value after a finite number of iterations of the function are known as periodic points; a fixed point is a periodic point with period equal to one. In projective geometry, a fixed point of a collineation is called a double point.