increasing/decreasing/monotone function

DefinitionLet $A$ 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.
The set of points where $f$ is discontinuous is at mostcountable [1, 2].
Lebesgue
$f$ is differentiable almosteverywhere ([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.

单词	IncreasingdecreasingmonotoneFunction
释义	increasing/decreasing/monotone function DefinitionLet $A$ 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. The set of points where $f$ is discontinuous is at mostcountable [1, 2]. Lebesgue $f$ is differentiable almosteverywhere ([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.
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