Dictionary

monotone function en · NOUN

Meanings

1. 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).
   • 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:
   • 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:
   • Where defined, the first derivative of a monotone function never changes sign, although it may be zero.
   • 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:
2. A 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. (qualifier:Boolean algebra) A 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.

Forms

Hypernyms

function

Hyponyms

order-reversing function · monotonic nonincreasing function (function that never increases as its independent variable increases) · nondecreasing function (function that never decreases as its independent variable increases) · monotone nonincreasing function (function that never increases as its independent variable increases) · monotonically nondecreasing function (function that never decreases as its independent variable increases) · monotone nondecreasing function (function that never decreases as its independent variable increases) · monotone decreasing function (function that never increases as its independent variable increases) · antitone function · monotonic nondecreasing function (function that never decreases as its independent variable increases) · monotone increasing function (function that never decreases as its independent variable increases) · monotonically nonincreasing function (function that never increases as its independent variable increases) · nonincreasing function (function that never increases as its independent variable increases) · decreasing function (function that never increases as its independent variable increases) · monotonically decreasing function (function that never increases as its independent variable increases) · order-preserving function · increasing function (function that never decreases as its independent variable increases) · monotonic increasing function (function that never decreases as its independent variable increases) · monotonically increasing function (function that never decreases as its independent variable increases) · isotone function · monotonic decreasing function (function that never increases as its independent variable increases) · anti-monotone function

Relateds

monotone transformation · weakly decreasing · strictly increasing · monotonic · weakly increasing · absolutely monotonic · monotone · monotone operator · strictly decreasing

Synonyms

monotonic function (function that either never decreases or never increases)

Translations (13)

pl funkcja monotoniczna (calculus: real function that either never decreases or never increases) · sh monotona funkcija (calculus: real function that either never decreases or never increases) · sh monotona funkcija (order theory: function such that either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)) · tr monoton fonksiyon (order theory: function such that either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)) · es función monótona (order theory: function such that either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)) · de monotone Funktion (calculus: real function that either never decreases or never increases) · de monotone Funktion (order theory: function such that either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)) · tr monoton fonksiyon (calculus: real function that either never decreases or never increases) · pl funkcja monotoniczna (order theory: function such that either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x)) · pl funkcja monotoniczna (Boolean function such that switching an input from 0 to 1 either does not change the output or changes it from 0 to 1) · bg монотонна функция (Boolean function such that switching an input from 0 to 1 either does not change the output or changes it from 0 to 1) · bg монотонна функция (calculus: real function that either never decreases or never increases) · es función monótona (calculus: real function that either never decreases or never increases)