# 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. 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. 2.

$f$ is strictly increasing or strongly increasing, if $x implies that $f(x).

3. 3.

$f$ is decreasing or weakly decreasing, if $x\leq y$ implies that $f(x)\geq f(y)$.

4. 4.

$f$ is strictly decreasing or strongly decreasing if $x implies that $f(x)>f(y)$.

5. 5.

$f$ is monotone, if $f$ is either increasing or decreasing.

6. 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. 1.

The set of points where $f$ is discontinuous is at most countable [1, 2].

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