Monotone function
A function of one variable, defined on a subset of the real numbers, whose increment $ \Delta f ( x) = f ( x ^ \prime )  f ( x) $,
for $ \Delta x = x ^ \prime  x > 0 $,
does not change sign, that is, is either always negative or always positive. If $ \Delta f ( x) $
is strictly greater (less) than zero when $ \Delta x > 0 $,
then the function is called strictly monotone (see Increasing function; Decreasing function). The various types of monotone functions are represented in the following table.
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If at each point of an interval $ f $ has a derivative that does not change sign (respectively, is of constant sign), then $ f $ is monotone (strictly monotone) on this interval.
The idea of a monotone function can be generalized to functions of various classes. For example, a function $ f ( x _ {1} \dots x _ {n} ) $ defined on $ \mathbf R ^ {n} $ is called monotone if the condition $ x _ {1} \leq x _ {1} ^ \prime \dots x _ {n} \leq x _ {n} ^ \prime $ implies that everywhere either $ f ( x _ {1} \dots x _ {n} ) \leq f ( x _ {1} ^ \prime \dots x _ {n} ^ \prime ) $ or $ f ( x _ {1} \dots x _ {n} ) \geq f ( x _ {1} ^ \prime \dots x _ {n} ^ \prime ) $ everywhere. A monotone function in the algebra of logic is defined similarly.
A monotone function of many variables, increasing or decreasing at some point, is defined as follows. Let $ f $ be defined on the $ n $ dimensional closed cube $ Q ^ {n} $, let $ x _ {0} \in Q ^ {n} $ and let $ E _ {t} = \{ {x } : {f ( x) = t, x \in Q ^ {n} } \} $ be a level set of $ f $. The function $ f $ is called increasing (respectively, decreasing) at $ x _ {0} $ if for any $ t $ and any $ x ^ \prime \in Q ^ {n} \setminus E _ {t} $ not separated in $ Q ^ {n} $ by $ E _ {t} $ from $ x _ {0} $, the relation $ f ( x ^ \prime ) < t $( respectively, $ f ( x ^ \prime ) > t $) holds, and for any $ x ^ {\prime\prime} \in Q ^ {n} \setminus E _ {t} $ that is separated in $ Q ^ {n} $ by $ E _ {t} $ from $ x _ {0} $, the relation $ f ( x ^ {\prime\prime} ) > t $( respectively, $ f ( x ^ {\prime\prime} ) < t $) holds. A function that is increasing or decreasing at some point is called monotone at that point.
Comments
For the concept in nonlinear functional analysis, see Monotone operator. For the concept in general partially ordered sets, see Monotone mapping.
Monotone function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monotone_function&oldid=47894