# Monotonic Function

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## monotonic function

[¦män·ə¦tän·ik ′fəŋk·shən]*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Monotonic Function

(or monotone function), a function whose increments Δ*f*(*x*) = *f*(*x*′) − *f*(*x*) do not change sign when Δ*x* = *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* = *x*^{3} 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.

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* = 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 *X*_{0}:

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