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.