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 range are the same.