increasing/decreasing/monotone function
DefinitionLet be a subset of , and let be a function from .Then
- 1.
is increasing or weakly increasing, if implies that (for all and in ).
- 2.
is strictly increasing or strongly increasing, if implies that .
- 3.
is decreasing or weakly decreasing, if implies that .
- 4.
is strictly decreasing or strongly decreasing if implies that .
- 5.
is monotone
,if is either increasing or decreasing.
- 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.
- 1.
The set of points where is discontinuous
is at mostcountable
[1, 2].
- Lebesgue
is differentiable
almosteverywhere ([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.