Functionsremain

Functionsremain is a term used in some discussions of mathematics and programming to denote the subset of a function's outputs that persists under a given set of transformations or iterations. It is not a standard term with a universal formal definition, and its precise meaning varies by author and field. In practice, it is used to discuss stability, invariance, or convergence of function values.

In dynamical systems and fixed-point theory, a common interpretation identifies remains with limit points of an orbit. Given a function f: X→X and a starting point x∈X, the omega-limit set of x consists of all y that are limit points of the sequence f^n(x). Some writers describe this long-run set as the remain of x under f, and the union over x in a subset of X yields the remain set of the system. This usage is informal but captures the idea of values that survive repeated application.

In functional programming and program analysis, remains are sometimes discussed as outputs that are preserved under refactoring, optimization, or partial evaluation. For example, when composing higher-order functions, the remain set may refer to results that do not change under a given transformation class, aiding reasoning about referential transparency and correctness.

An example is the fixed point of the cosine function, approximately y ≈ 0.739085... that satisfies cos(y) = y. Under iteration from any starting value, the orbit approaches this y, which can be described as the remain of the system.

See also fixed point, omega-limit set, and invariant.