monotone function
noun1
Calculus, Mathematics, Sciences 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.
- 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:
2 more examples
- 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:
2
Mathematical analysis, Mathematics, Order theory, Sciences 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
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.
Boolean algebra
- Synonyms
- monotonic function
- Related terms
- monotone, monotone operator, monotone transformation, monotonic, absolutely monotonic, strictly decreasing, strictly increasing, weakly decreasing, weakly increasing
- Hypernyms
- function
- Hyponyms
- nondecreasing function, monotone nondecreasing function, monotonic nondecreasing function, monotonically nondecreasing function, increasing function, monotone increasing function, monotonic increasing function, monotonically increasing function, nonincreasing function, monotone nonincreasing function, monotonic nonincreasing function, monotonically nonincreasing function, decreasing function, monotone decreasing function, monotonic decreasing function, monotonically decreasing function, isotone function, order-preserving function, anti-monotone function, antitone function, order-reversing function