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increasing/decreasing/monotone function
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(Definition)
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Definition Let  be a subset of
 , and let  be a function from
 . Then
 is increasing or weakly increasing, if  implies that
 (for all  and  in  ). is strictly increasing or strongly increasing, if  implies that
 . is decreasing or weakly decreasing, if  implies that
 . is strictly decreasing or strongly decreasing if  implies that
 . is monotone, if  is either increasing or decreasing. is strictly monotone or strongly monotone, if  is either strictly increasing or strictly decreasing.
Theorem Let  be a bounded or unbounded open interval of
 . In other words, let  be an interval of the form  , where
 . Futher, let
 be a monotone function.
- The set of points where
 is discontinuous is at most countable [1,2].
- [Lebesgue]
 is differentiable almost everywhere ([3], pp. 514).
- 1
- C.D. Aliprantis, O. Burkinshaw, Principles of Real Analysis, 2nd ed., Academic Press, 1990.
- 2
- W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Inc., 1976.
- 3
- F. Jones, Lebesgue Integration on Euclidean Spaces, Jones and Barlett Publishers, 1993.
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"increasing/decreasing/monotone function" is owned by Koro. [ full author list (2) | owner history (1) ]
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(view preamble)
| Also defines: |
increasing, decreasing, strictly increasing, strictly decreasing, monotone, monotonic, strictly monotone, strictly monotonic, weakly increasing, weakly decreasing, strongly increasing, strongly decreasing, strongly monotone, weakly monotone, stronly monotonic, weakly monotonic |
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Cross-references: almost everywhere, differentiable, countable, discontinuous, points, monotone function, interval, open interval, unbounded, bounded, implies, function, subset
There are 130 references to this entry.
This is version 9 of increasing/decreasing/monotone function, born on 2003-04-28, modified 2008-03-27.
Object id is 4228, canonical name is IncreasingdecreasingmonotoneFunction.
Accessed 27566 times total.
Classification:
| AMS MSC: | 26A06 (Real functions :: Functions of one variable :: One-variable calculus) | | | 26A48 (Real functions :: Functions of one variable :: Monotonic functions, generalizations) |
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Pending Errata and Addenda
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