Monotone Functions

Figure 1. A monotonically non-decreasing function.

Figure 2. A monotonically non-increasing function

Figure 3. A function that is not monotonic

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.

In calculus and analysis

In calculus, a function $f$ defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease.

A function is called monotonically increasing (also increasing or non-decreasing), if for all $x$ and $y$ such that $x\leq y$ one has $f\!\left(x\right)\leq f\!\left(y\right)$, so $f$ preserves the order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasing or non-increasing) if, whenever $x\leq y$, then $f\!\left(x\right)\geq f\!\left(y\right)$, so it reverses the order (see Figure 2).

If the order $\leq$ in the definition of monotonicity is replaced by the strict order $<$, then one obtains a stronger requirement. A function with this property is called strictly increasing (also increasing). Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing (also decreasing). A function may be called strictly monotone if it is either strictly increasing or strictly decreasing. Functions that are strictly monotone are one-to-one (because for $x$ not equal to $y$, either $x<y$ or $x>y$ and so, by monotonicity, either $f\!\left(x\right)<f\!\left(y\right)$ or $f\!\left(x\right)>f\!\left(y\right)$, thus $f\!\left(x\right)\neq f\!\left(y\right)$.)

If it is not clear that "increasing" and "decreasing" are taken to include the possibility of repeating the same value at successive arguments, one may use the terms weakly monotone, weakly increasing and weakly decreasing to stress this possibility.

The terms "non-decreasing" and "non-increasing" should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing". For example, the function of figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not increasing, but it is neither non-decreasing nor non-increasing.

A function $f\!\left(x\right)$ is said to be absolutely monotonic over an interval $\left(a,b\right)$ if the derivatives of all orders of $f$ are nonnegative or all nonpositive at all points on the interval.

Inverse of function

A function that is monotonic, but not strictly monotonic, and thus constant on an interval, doesn't have an inverse. This is because in order for a function to have an inverse, there needs to be a one-to-one mapping from the range to the domain of the function. Since a monotonic function has some values that are constant in its domain, this means that there would be more than one value in the range that maps to this constant value.

However, a function y = g(x) that is strictly monotonic, has an inverse function such that x = h(y) because there is guaranteed to always be a one-to-one mapping from range to domain of the function. Also, a function can be said to be strictly monotonic on a range of values, and thus have an inverse on that range of value. For example, if y = g(x) is strictly monotonic on the range [a,b], then it has an inverse x = h(y) on the range [g(a), g(b)], but we cannot say the entire range of the function has an inverse.

Note, some textbooks mistakenly state that an inverse exists for a monotonic function, when they really mean that an inverse exists for a strictly monotonic function.

Monotonic transformation

The term monotonic transformation (or monotone transformation) can also possibly cause some confusion because it refers to a transformation by a strictly increasing function. This is the case in economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform (see also monotone preferences). In this context, what we are calling a "monotonic transformation" is, more accurately, called a "positive monotonic transformation", in order to distinguish it from a “negative monotonic transformation,” which reverses the order of the numbers.

Some basic applications and results

The following properties are true for a monotonic function $f\colon \mathbb {R} \to \mathbb {R}$:

• $f$ has limits from the right and from the left at every point of its domain;
• $f$ has a limit at positive or negative infinity ($\pm \infty$) of either a real number, $\infty$, or $-\infty$.
• $f$ can only have jump discontinuities;
• $f$ can only have countably many discontinuities in its domain. The discontinuities, however, do not necessarily consist of isolated points and may even be dense in an interval (a, b).}}

These properties are the reason why monotonic functions are useful in technical work in analysis. Some more facts about these functions are:

• if $f$ is a monotonic function defined on an interval $I$, then $f$ is differentiable almost everywhere on $I$; i.e. the set of numbers $x$ in $I$ such that $f$ is not differentiable in $x$ has Lebesgue measure zero. In addition, this result cannot be improved to countable: see Cantor function.
• if this set is countable, then $f$ is absolutely continuous.
• if $f$ is a monotonic function defined on an interval $\left[a,b\right]$, then $f$ is Riemann integrable.

An important application of monotonic functions is in probability theory. If $X$ is a random variable, its cumulative distribution function $F_{X}\!\left(x\right)={\text{Prob}}\!\left(X\leq x\right)$ is a monotonically increasing function.

A function is unimodal if it is monotonically increasing up to some point (the mode) and then monotonically decreasing.

When $f$ is a strictly monotonic function, then $f$ is injective on its domain, and if $T$ is the range of $f$, then there is an inverse function on $T$ for $f$. In contrast, each constant function is monotonic, but not injective, and hence cannot have an inverse.

In topology

A map $f:X\rightarrow Y$ is said to be monotone if each of its fibers is connected; i.e. for each element $y$ in $Y$ the (possibly empty) set $f^{-1}(y)$ is connected.

In functional analysis

In functional analysis on a topological vector space $X$, a (possibly non-linear) operator $T:X\rightarrow X^{*}$ is said to be a monotone operator if

$(Tu-Tv,u-v)\geq 0\quad \forall u,v\in X.$

Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives.

A subset $G$ of $X\times X^{*}$ is said to be a monotone set if for every pair $[u_{1},w_{1}]$ and $[u_{2},w_{2}]$ in $G$,

$(w_{1}-w_{2},u_{1}-u_{2})\geq 0.$

$G$ is said to be maximal monotone if it is maximal among all monotone sets in the sense of set inclusion. The graph of a monotone operator $G(T)$ is a monotone set. A monotone operator is said to be maximal monotone if its graph is a maximal monotone set.

Resources

• Monotonic function, Wikipedia