In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically, for functions, a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an invariant set.

Formally, c is a fixed point of a function f if c belongs to both the domain and the codomain of f, and f(c) = c. In particular, f cannot have any fixed point if its domain is disjoint from its codomain. If f is defined on the real numbers, it corresponds, in graphical terms, to a curve in the Euclidean plane, and each fixed-point c corresponds to an intersection of the curve with the line y = x, cf. picture.

For example, if f is defined on the real numbers by ${\displaystyle f(x)=x^{2}-3x+4,}$ then 2 is a fixed point of f, because f(2) = 2.

Not all functions have fixed points: for example, f(x) = x + 1 has no fixed points because x + 1 is never equal to x for any real number.

In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. Specifically, given a function ${\displaystyle f}$ with the same domain and codomain, a point ${\displaystyle x_{0}}$ in the domain of ${\displaystyle f}$, the fixed-point iteration is

$${\displaystyle x_{n+1}=f(x_{n}),\,n=0,1,2,\dots }$$

which gives rise to the sequence ${\displaystyle x_{0},x_{1},x_{2},\dots }$ of iterated function applications ${\displaystyle x_{0},f(x_{0}),f(f(x_{0})),\dots }$ which is hoped to converge to a point ${\displaystyle x}$. If ${\displaystyle f}$ is continuous, then one can prove that the obtained ${\displaystyle x}$ is a fixed point of ${\displaystyle f}$.

The notions of attracting fixed points, repelling fixed points, and periodic points are defined with respect to fixed-point iteration.