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.

<tbody> </tbody> $ \Delta f ( x) \geq 0 $ Increasing (non-decreasing) $ \Delta f ( x) \leq 0 $ Decreasing (non-increasing) $ \Delta f ( x) > 0 $ Strictly increasing $ \Delta f ( x) < 0 $ Strictly decreasing

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 non-linear functional analysis, see Monotone operator. For the concept in general partially ordered sets, see Monotone mapping.