increasing/decreasing/monotone function
Definition Let A be a subset of ℝ, and let f be a function from f:A→ℝ. Then
-
1.
f is increasing or weakly increasing, if x≤y implies that f(x)≤f(y) (for all x and y in A).
-
2.
f is strictly increasing or strongly increasing, if x<y implies that f(x)<f(y).
-
3.
f is decreasing or weakly decreasing, if x≤y implies that f(x)≥f(y).
-
4.
f is strictly decreasing or strongly decreasing if x<y implies that f(x)>f(y).
-
5.
f is monotone
, if f is either increasing or decreasing.
-
6.
f is strictly monotone or strongly monotone, if f is either strictly increasing or strictly decreasing.
Theorem Let X be a bounded or unbounded
open interval of ℝ.
In other words, let X be an interval of the form X=(a,b), where a,b∈ℝ∪{-∞,∞}.
Futher, let f:X→ℝ be a monotone function.
-
1.
The set of points where f is discontinuous
is at most countable
[1, 2].
-
Lebesgue
f 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 |