increasing/decreasing/monotone function
Definition Let $A$ be a subset of $\mathbb{R}$, and let $f$ be a function from $f:A\to \mathbb{R}$. Then

1.
$f$ is increasing or weakly increasing, if $x\le y$ implies that $f(x)\le f(y)$ (for all $x$ and $y$ in $A$).

2.
$f$ is strictly increasing or strongly increasing, if $$ implies that $$.

3.
$f$ is decreasing or weakly decreasing, if $x\le y$ implies that $f(x)\ge f(y)$.

4.
$f$ is strictly decreasing or strongly decreasing if $$ 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 $\mathbb{R}$. In other words, let $X$ be an interval of the form $X=(a,b)$, where $a,b\in \mathbb{R}\cup \{\mathrm{\infty},\mathrm{\infty}\}$. Futher, let $f:X\to \mathbb{R}$ 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, McGrawHill 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  20130322 13:36:05 
Last modified on  20130322 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 