increasing/decreasing/monotone function
Definition Let be a subset of , and let be a function from . Then
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1.
is increasing or weakly increasing, if implies that (for all and in ).
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2.
is strictly increasing or strongly increasing, if implies that .
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3.
is decreasing or weakly decreasing, if implies that .
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4.
is strictly decreasing or strongly decreasing if implies that .
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5.
is monotone, if is either increasing or decreasing.
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6.
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.
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1.
The set of points where is discontinuous is at most countable [1, 2].
-
Lebesgue
is differentiable almost everywhere ([3], pp. 514).
References
- 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.
Title | increasing/decreasing/monotone function |
Canonical name | IncreasingdecreasingmonotoneFunction |
Date of creation | 2013-03-22 13:36:05 |
Last modified on | 2013-03-22 13:36:05 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 12 |
Author | Koro (127) |
Entry type | Definition |
Classification | msc 26A06 |
Classification | msc 26A48 |
Defines | increasing |
Defines | decreasing |
Defines | strictly increasing |
Defines | strictly decreasing |
Defines | monotone |
Defines | monotonic |
Defines | strictly monotone |
Defines | strictly monotonic |
Defines | weakly increasing |
Defines | weakly decreasing |
Defines | strongly increasing |
Defines | strongly decreasing |
Defines | strongly monotone |
Defines | weakly monotone |
Defines | stronly mono |