## You are here

Homeincreasing/decreasing/monotone function

## Primary tabs

# 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 $x\leq y$ implies that $f(x)\leq f(y)$ (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 $x\leq y$ implies that $f(x)\geq f(y)$.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 $\mathbb{R}$. In other words, let $X$ be an interval of the form $X=(a,b)$, where $a,b\in\mathbb{R}\cup\{-\infty,\infty\}$. Futher, let $f:X\to\mathbb{R}$ be a monotone function.

1. - 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.

## Mathematics Subject Classification

26A06*no label found*26A48

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Prove that for any sets A, B, and C, An(BUC)=(AnB)U(AnC) by St_Louis

Apr 20

new image: information-theoretic-distributed-measurement-dds.png by rspuzio

new image: information-theoretic-distributed-measurement-4.2 by rspuzio

new image: information-theoretic-distributed-measurement-4.1 by rspuzio

new image: information-theoretic-distributed-measurement-3.2 by rspuzio

new image: information-theoretic-distributed-measurement-3.1 by rspuzio

new image: information-theoretic-distributed-measurement-2.1 by rspuzio

Apr 19

new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

## Attached Articles

## Corrections

minor typo by neapol1s ✓

missing word by Mathprof ✓

synonym by CWoo ✓

spelling by CWoo ✓

## Comments

## Question on Definition.

Let set A = {1,2,3}.

1. How many relations are monotone increasing funtions?

2. How many relations are monotone decreasing funtions?

3. How many relations are strictly increasing funtions?