Monotonic Function

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monotonic function

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Monotonic Function


(or monotone function), a function whose increments Δf(x) = f(x′) − f(x) do not change sign when Δx = x′ − x > 0; that is, the increments are either always nonnegative or always nonpositive. Somewhat inaccurately, a monotonic function can be defined as a function that always varies in the same direction. Different types of monotonic functions are represented in Figure 1. For example, the function y = x3 is an increasing function. If a function f(x) has a derivative f′(x) that is nonnegative at every point and that vanishes only at a finite number of individual points, then f(x) is an increasing function. Similarly, if f′(x) ≤ 0 and vanishes only at a finite number of points, then f(x) is a decreasing function.

Figure 1

A monotonicity condition can hold either for all x or for x on a given interval. In the latter case, the function is said to be monotonic on this interval. For example, the function y = Monotonic Function increases on the interval [−1,0] and decreases on the interval [0, +1]. A monotonic function is one of the simplest classes of functions and is continually encountered in mathematical analysis and the theory of functions. If f(x) is a monotonic function, then the following limits exist for any X0:


References in periodicals archive ?
As the optimal price is invariant under a monotone transformation, we may equivalently consider the objective
This illustrates the possibility that applying a judicious monotone transformation to an objective function may change the comparative statics analysis.
Because the exponential (exp) is a monotone transformation and exp(log) is the identity function, it follows that [PI] also satisfies the single-crossing property for all D functions because the latter property is preserved by monotone transformations.
9) Observe here that taking monotone transformation of a nonsupermodular objective function may bring about supermolodularity of the transformed objective.