monotone function
Language: English
Part of speech: noun
Forms: form: monotone functions
form: monotonic function
Senses: examples: text: Where defined, the first derivative of a monotone function never changes sign, although it may be zero.
text: Section 1 contains Lebesgue's main tool, a theorem saying that monotone functions on the line are differentiable almost everywhere.
ref: 2005, Anthony W. Knapp, Basic Real Analysis, Springer, page 357:
text: 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;.
ref: 2011, Saminathan Ponnusamy, Foundations of Mathematical Analysis, Springer, page 469:
text: 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).
ref: 2013, Donald Yau, A First Course in Analysis, World Scientific, page 104:
topics: calculus
mathematics
sciences
glosses: 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).
topics: mathematical-analysis
mathematics
order-theory
sciences
glosses: 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).
qualifier: Boolean algebra
glosses: 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.
Synonyms: sense: function that either never decreases or never increases
word: monotonic function
Hypernyms: function
Hyponyms: sense: function that never decreases as its independent variable increases
word: nondecreasing function
sense: function that never decreases as its independent variable increases
word: monotone nondecreasing function
sense: function that never decreases as its independent variable increases
word: monotonic nondecreasing function
sense: function that never decreases as its independent variable increases
word: monotonically nondecreasing function
sense: function that never decreases as its independent variable increases
word: increasing function
sense: function that never decreases as its independent variable increases
word: monotone increasing function
sense: function that never decreases as its independent variable increases
word: monotonic increasing function
sense: function that never decreases as its independent variable increases
word: monotonically increasing function
sense: function that never increases as its independent variable increases
word: nonincreasing function
sense: function that never increases as its independent variable increases
word: monotone nonincreasing function
sense: function that never increases as its independent variable increases
word: monotonic nonincreasing function
sense: function that never increases as its independent variable increases
word: monotonically nonincreasing function
sense: function that never increases as its independent variable increases
word: decreasing function
sense: function that never increases as its independent variable increases
word: monotone decreasing function
sense: function that never increases as its independent variable increases
word: monotonic decreasing function
sense: function that never increases as its independent variable increases
word: monotonically decreasing function
topics: order-theory
mathematics
sciences
word: isotone function
topics: order-theory
mathematics
sciences
word: order-preserving function
topics: order-theory
mathematics
sciences
word: anti-monotone function
topics: order-theory
mathematics
sciences
word: antitone function
topics: order-theory
mathematics
sciences
word: order-reversing function
Related terms: monotone
monotone operator
monotone transformation
monotonic
absolutely monotonic
strictly decreasing
strictly increasing
weakly decreasing
weakly increasing