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A monotonically non-increasing function Figure 3. 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 differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope. [3] [4] Points where concavity changes (between concave and convex) are inflection points. [5]
Such an ambiguity can be mitigated by writing "x is strictly positive" for x > 0, and "x is non-negative" for x ≥ 0. (A precise term like non-negative is never used with the word negative in the wider sense that includes zero.) The word "proper" is often used in the same way as "strict".
Every supermodular function is quasisupermodular. As in the case of single crossing differences, and unlike supermodularity, quasisupermodularity is an ordinal property. That is, if function is quasisupermodular, then so is function :=, where is some strictly increasing function.
If a sequence is either increasing or decreasing it is called a monotone sequence. This is a special case of the more general notion of a monotonic function. The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoid any possible confusion with strictly increasing and strictly decreasing ...
If is (strictly) Schur-convex and is (strictly) monotonically increasing, then is (strictly) Schur-convex. If is a convex function defined on a real interval ...
Ordinal exponentiation is strictly increasing and continuous in the right argument: If γ > 1 and α < β, then γ α < γ β. If α < β, then α γ ≤ β γ. Note, for instance, that 2 < 3 and yet 2 ω = 3 ω = ω. If α > 1 and α β = α γ, then β = γ. If α = 1 or α = 0 this is not the case.
Strictly increasing in income; Homogenous with degree zero in prices and income; if prices and income are all multiplied by a given constant the same bundle of consumption represents a maximum, so optimal utility does not change; quasi-convex in (p,w).