Substantiv
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).
- Verwandte Wörter
- monotonemonotone operatormonotone transformationmonotonicabsolutely monotonicstrictly decreasingstrictly increasingweakly decreasingweakly increasing
- Oberbegriffe
- Unterbegriffe
- isotone functionorder-preserving functionanti-monotone functionantitone functionorder-reversing function
- Beispiele
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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.
Weitere Beispiele (2)
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;.
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).
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).
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.