monotone function

English

Alternative forms

• monotonic function

Noun

monotone function (plural monotone functions)

1. (calculus) 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.

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

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

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

     Monotone functions on ${\displaystyle }$ have nice properties. For example, they are integrable on ${\displaystyle }$ 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 ${\displaystyle {\mathit {BV}}()}$ contains the class of monotone functions on ${\displaystyle }$.

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

     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).

2. (order theory, mathematical analysis) 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. (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.

Usage notes

• The order theory definition avoids reference to the concepts increasing and decreasing, making it somewhat more generally applicable. Strictly speaking, the partial orders for X and Y need not be related (the notation "≤" is conventional). This case encompasses the possibility that X and Y are multidimensional spaces (e.g. Rⁿ) and f is a mapping between them.
• In the Boolean algebra case, there is implicit in the definition an intuitively natural partial order "≤" (see product order on Wikipedia) such that, given two input tuples a = (a₁, a₂,... a_n) and b = (b₁, b₂,... b_n), a ≤ b means that b can be obtained from a via a series of (zero or more) steps each switching an input from 0 to 1. With this partial order in mind, (only) property (1) of the order theory definition applies.

Synonyms

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

Hypernyms

• function

Hyponyms

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

Related terms

• monotone
• monotone operator
• monotone transformation
• monotonic

Translations

See also

• absolutely monotonic
• strictly decreasing
• strictly increasing
• weakly decreasing
• weakly increasing

Further reading

• Monotonic function on Wikipedia.Wikipedia
• Bernstein's theorem on monotone functions on Wikipedia.Wikipedia
• Monotone cubic interpolation on Wikipedia.Wikipedia
• Monotone function on Encyclopedia of Mathematics
• Monotonic Function on Wolfram MathWorld