increasing/decreasing/monotone function

Definition Let $A$ be a subset of $\sR$ , and let $f$ be a function from $f:A\to \sR$ . 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 $x< y$ implies that $f(x)< f(y)$ .
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 $x< y$ 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 $\sR$ . In other words, let $X$ be an interval of the form $X=(a,b)$ , where $a,b\in\sR\cup\{-\infty,\infty\}$ . Futher, let $f:X\to \sR$ be a monotone function.

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).

Bibliography

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.