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A function that is not monotonic. In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. [1] [2] [3] This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
A function that is absolutely monotonic on [,) can be extended to a function that is not only analytic on the real line but is even the restriction of an entire function to the real line. The big Bernshtein theorem : A function f ( x ) {\displaystyle f(x)} that is absolutely monotonic on ( − ∞ , 0 ] {\displaystyle (-\infty ,0]} can be ...
A contraction mapping has at most one fixed point. Moreover, the Banach fixed-point theorem states that every contraction mapping on a non-empty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point
A residuated function and its residual form a Galois connection under the (more recent) monotone definition of that concept, and for every (monotone) Galois connection the lower adjoint is residuated with the residual being the upper adjoint. [2] Therefore, the notions of monotone Galois connection and residuated mapping essentially coincide.
It can be shown (see Blyth or Erné for proofs) that a function f is a lower (respectively upper) adjoint if and only if f is a residuated mapping (respectively residual mapping). Therefore, the notion of residuated mapping and monotone Galois connection are essentially the same.
In order theory, a branch of mathematics, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order embeddings constitute a notion which is strictly weaker than the concept of an order isomorphism.
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. [1] [2] [3] That is, a function : is open if for any open set in , the image is open in . Likewise, a closed map is a function that maps closed sets to closed sets.
Kneading theory provides an effective calculus for describing the qualitative behavior of the iterates of a piecewise monotone mapping f of a closed interval I of the real line into itself. Some quantitative invariants of this discrete dynamical system , such as the lap numbers of the iterates and the Artin–Mazur zeta function of f are ...