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# 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*

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## Corrections

minor typo by neapol1s ✓

missing word by Mathprof ✓

synonym by CWoo ✓

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## 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?