monotone function

noun

A function f : X→R (where X is a subset of R, possibly a discrete set) that either never decreases or never increases as its independent variable increases; that is, either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x).

“Where defined, the first derivative of a monotone function never changes sign, although it may be zero.”

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monotonic function
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Meanings

nounmonotone function (plural monotone functions)

plural monotone functions · alternative monotonic function

  1. calculusA function f : X→R (where X is a subset of R, possibly a discrete set) that either never decreases or never increases as its independent variable increases; that is, either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x).

    Where defined, the first derivative of a monotone function never changes sign, although it may be zero.

    3 more examples

    Section 1 contains Lebesgue's main tool, a theorem saying that monotone functions on the line are differentiable almost everywhere.

    2005, Anthony W. Knapp, Basic Real Analysis, Springer, page 357

    Monotone functions on #91;a,b#93; have nice properties. For example, they are integrable on #91;a,b#93; and have only a countable number of jump discontinuities. In this section, we shall also show that every monotone function is a function of bounded variation, and hence the class #92;mathit#123;BV#125;(#91;a,b#93;) contains the class of monotone functions on #91;a,b#93;.

    2011, Saminathan Ponnusamy, Foundations of Mathematical Analysis, Springer, page 469

    We saw in the previous section that monotone functions have some nice properties. For example, a monotone function is continuous except possibly on a countable set. They are also closed under scalar multiplication, and the sum of two increasing functions is increasing. (Exercise (4) on page 103). However, the difference and product of two monotone functions are not necessarily monotone (Exercise (5) on page 103).

    2013, Donald Yau, A First Course in Analysis, World Scientific, page 104

    Synonyms: monotonic function

  2. mathematical analysisA function f : X→Y (where X and Y are posets with partial order "≤") with either: (1) the property that x ≤ y implies f(x) ≤ f(y), or (2) the property that x ≤ y implies f(y) ≤ f(x).
  3. Boolean algebraA Boolean function with the property that switching any one input variable from 0 to 1 results either in no change in output or a change from 0 to 1.

Synonyms & antonyms

nounA function f : X→R (where X is a subset of R, possibly a discrete set) that either never decreases or never increases as its independent variable increases; that is, either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x).

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monotonic function

Translations

Bulgarian монотонна функция, монотонна функция
  • монотонна функция (monotonna funkcija)calculus: real function that either never decreases or never increases
  • монотонна функция (monotonna funkcija)Boolean function such that switching an input from 0 to 1 either does not change the output or changes it from 0 to 1
German monotone Funktion, monotone Funktion
  • monotone Funktioncalculus: real function that either never decreases or never increases
  • monotone Funktionorder theory: function such that either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)
Polish funkcja monotoniczna, funkcja monotoniczna
  • funkcja monotonicznacalculus: real function that either never decreases or never increases
  • funkcja monotonicznaBoolean function such that switching an input from 0 to 1 either does not change the output or changes it from 0 to 1
  • funkcja monotonicznaorder theory: function such that either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)
Spanish función monótona, función monótona
  • función monótonacalculus: real function that either never decreases or never increases
  • función monótonaorder theory: function such that either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)
Turkish monoton fonksiyon, monoton fonksiyon
  • monoton fonksiyoncalculus: real function that either never decreases or never increases
  • monoton fonksiyonorder theory: function such that either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)
Cite this entry
APA

monotone function. (n.d.). In WatkinsX Dictionary. Retrieved November 2, 2026, from https://dictionary.watkinsx.com/word/monotone%20function

MLA

“monotone function.” WatkinsX Dictionary, https://dictionary.watkinsx.com/word/monotone%20function. Accessed November 2, 2026.

Chicago

WatkinsX Dictionary, s.v. “monotone function,” accessed November 2, 2026, https://dictionary.watkinsx.com/word/monotone%20function.

Source: monotone function on Wiktionary, CC BY-SA 4.0.