Monotonic Function

A Monotonic Function is a function that has an ordered function output.

• AKA: Monotone Function.
• Context:
  • It can range from being a Monotone Increasing Function to being a Monotone Nondecreasing Function to being a Monotone Increasing Function to being a Monotone Increasing Function.
• Example(s):
  • a Cumulative Distribution Function.
  • a Monotonic Boolean Function.
  • …
• Counter-Example(s):
  • a Non-Monotonic Function.
• See: Monotonic, Modular Function, Partially Ordered Set.

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References

2017

• (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/monotonic_function Retrieved:2017-8-14.
  • 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.

2010

• http://img.tfd.com/ggse/2c/gsed_0001_0016_0_img4169.png
  • QUOTE:

2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Monotonic_function
  • … In calculus, a function [math]\displaystyle{ f }[/math] defined on a subset of the real numbers with real values is called monotonic (also monotonically increasing or non-decreasing), if for all [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] such that [math]\displaystyle{ x }[/math] ≤ [math]\displaystyle{ y }[/math] one has [math]\displaystyle{ f }[/math](x) ≤ [math]\displaystyle{ f }[/math](y), so [math]\displaystyle{ f }[/math] preserves the order (see Figure 1). Likewise, a function is called monotonically decreasing (non-increasing) if, whenever [math]\displaystyle{ x }[/math] ≤ [math]\displaystyle{ y }[/math], then [math]\displaystyle{ f }[/math](x) ≥ [math]\displaystyle{ f }[/math](y), so it reverses the order (see Figure 2).
  • The following properties are true for a monotonic function [math]\displaystyle{ f }[/math] : R → R:
• f has limits from the right and from the left at every point of its domain;
• f has a limit at infinity (either ∞ or −∞) of either a real number, ∞, or −∞.
• f can only have jump discontinuities;
• f can only have countably many discontinuities in its domain.
  • These properties are the reason why monotonic functions are useful in technical work in analysis. Two facts about these functions are:
• if [math]\displaystyle{ f }[/math] is a monotonic function defined on an interval $I$, then [math]\displaystyle{ f }[/math] is differentiable almost everywhere on $I$, i.e. the set of numbers [math]\displaystyle{ x }[/math] in $I$ such that [math]\displaystyle{ f }[/math] is not differentiable in [math]\displaystyle{ x }[/math] has Lebesgue measure zero.
• if [math]\displaystyle{ f }[/math] is a monotonic function defined on an interval [a, b], then [math]\displaystyle{ f }[/math] is Riemann integrable.
  • An important application of monotonic functions is in probability theory. If [math]\displaystyle{ X }[/math] is a random variable, its cumulative distribution function

F_X(x) = Prob(X ≤ x)

is a monotonically increasing function.

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

2006

• http://glossary.computing.society.informs.org/index.php?page=M.html
  • QUOTE: Monotonic function. A function that either never decreases or never increases. A non-decreasing, or isotonic, function satisfies: f(x') >= f(x) whenever x' >= x (it is strictly increasing if f(x') > f(x) for x' not= x). A non-increasing, or anatonic, function satisfies: f(x') <= f(x) whenever x' >= x (it is strictly decreasing if f(x') < f(x) for x' not= x). This extends to a vector function, where range(f) is in Rn: (f(x)-f(y))t (x-y) >= 0.

2002

• http://everything2.com/title/monotone+function

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