Definition:Fixed Point

Definition
Let $f: S \to T$ be a mapping.

Then a fixed point (or fixed element) of $S$ under $f$ is a point $x \in S$ such that $f \left({x}\right) = x$.

(It follows trivially that for $f$ to have any fixed points at all, it is necessary that $S \cap T \ne \varnothing$.)

The concept is usually encountered in the context of transformations, that is, a mapping where the domain and codomain are the same.