Darboux integral - Revision history

InfernalAtom683 at 11:09, 10 April 2026

2026-04-10T11:09:38Z

← Older revision		Revision as of 19:09, 10 April 2026
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	{{Terminology_table\|		{{Terminology_table\|
	{{Terminology_table/row \| Darboux integral \| intégrale de Darboux \| Darboux-Integral \| Darboux 积分 \| Darboux 積分 \| ダルブー積分 }}		{{Terminology_table/row \| Darboux integral \| intégrale de Darboux \| Darboux-Integral \| Darboux 积分 \| Darboux 積分 \| ダルブー積分 }}
			{{Terminology_table/row \| partition \| partition \| Partition \| 分割 \| 分割 \| 分割 }}
	{{Terminology_table/row \| Darboux sum \| somme de Darboux \| Darboux-Summe \| Darboux 和 \| Darboux 和 \| ダルブー和 }}		{{Terminology_table/row \| Darboux sum \| somme de Darboux \| Darboux-Summe \| Darboux 和 \| Darboux 和 \| ダルブー和 }}
	{{Terminology_table/row \| upper Darboux sum \| somme supérieure de Darboux \| obere Darboux-Summe \| 上 Darboux 和 \| 上 Darboux 和 \| 上ダルブー和 }}		{{Terminology_table/row \| upper Darboux sum \| somme supérieure de Darboux \| obere Darboux-Summe \| 上 Darboux 和 \| 上 Darboux 和 \| 上ダルブー和 }}

InfernalAtom683 at 11:08, 10 April 2026

2026-04-10T11:08:22Z

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	For each subinterval <math>[x_{i-1},x_i]</math> of <math>P</math>, define:		For each subinterval <math>[x_{i-1},x_i]</math> of <math>P</math>, define:

	* the infimum:		*the infimum:

	<math display="block">		<math display="block">
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	</math>		</math>

	* the supremum:		*the supremum:

	<math display="block">		<math display="block">
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	{{Terminology_table/row \| upper integral \| intégrale supérieure \| oberes Integral \| 上 Darboux 积分 \| 上 Darboux 積分 \| 上ダルブー積分 }}		{{Terminology_table/row \| upper integral \| intégrale supérieure \| oberes Integral \| 上 Darboux 积分 \| 上 Darboux 積分 \| 上ダルブー積分 }}
	{{Terminology_table/row \| lower integral \| intégrale inférieure \| unteres Integral \| 下 Darboux 积分 \| 下 Darboux 積分 \| 下ダルブー積分 }}		{{Terminology_table/row \| lower integral \| intégrale inférieure \| unteres Integral \| 下 Darboux 积分 \| 下 Darboux 積分 \| 下ダルブー積分 }}
	}}
	}}		}}

InfernalAtom683 at 11:07, 10 April 2026

2026-04-10T11:07:57Z

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	* [[Riemann integral]]		* [[Riemann integral]]
	* [[Partition]]		* [[Partition]]

			== Terminology ==
			{{Terminology_table\|
			{{Terminology_table/row \| Darboux integral \| intégrale de Darboux \| Darboux-Integral \| Darboux 积分 \| Darboux 積分 \| ダルブー積分 }}
			{{Terminology_table/row \| Darboux sum \| somme de Darboux \| Darboux-Summe \| Darboux 和 \| Darboux 和 \| ダルブー和 }}
			{{Terminology_table/row \| upper Darboux sum \| somme supérieure de Darboux \| obere Darboux-Summe \| 上 Darboux 和 \| 上 Darboux 和 \| 上ダルブー和 }}
			{{Terminology_table/row \| lower Darboux sum \| somme inférieure de Darboux \| untere Darboux-Summe \| 下 Darboux 和 \| 下 Darboux 和 \| 下ダルブー和 }}
			{{Terminology_table/row \| upper integral \| intégrale supérieure \| oberes Integral \| 上 Darboux 积分 \| 上 Darboux 積分 \| 上ダルブー積分 }}
			{{Terminology_table/row \| lower integral \| intégrale inférieure \| unteres Integral \| 下 Darboux 积分 \| 下 Darboux 積分 \| 下ダルブー積分 }}
			}}
			}}

InfernalAtom683 at 07:50, 9 April 2026

2026-04-09T07:50:02Z

← Older revision		Revision as of 15:50, 9 April 2026
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	<math display="block">\int_a^b f=\underline{\int_a^b}f=\overline{\int_a^b}f.</math>		<math display="block">\int_a^b f=\underline{\int_a^b}f=\overline{\int_a^b}f.</math>
	In this case, function <math>f</math> is said to be '''Darboux-integrable'''.		In this case, function <math>f</math> is said to be '''Darboux-integrable'''.

			== See also ==

			* [[Riemann integral]]
			* [[Partition]]

InfernalAtom683: Created page with "The '''Darboux integral''' is a formulation of integration in real analysis defined using upper and lower sums over partitions of an interval. It provides an order-theoretic approach to integration and is equivalent to the Riemann integral. == Definition == === Darboux sums === Let $f:[a,b]\to\mathbb{R}$ be a bounded function. Let $P=\{x_0,x_1,\dots,x_n\}, \quad a=x_0 < x_1 < \cdots < x_n=b$ be a partition of the interval..."

2026-03-28T07:29:39Z

Created page with "The '''Darboux integral''' is a formulation of integration in real analysis defined using upper and lower sums over partitions of an interval. It provides an order-theoretic approach to integration and is equivalent to the Riemann integral. == Definition == === Darboux sums === Let <math>f:[a,b]\to\mathbb{R}</math> be a bounded function. Let <math display="block"> P=\{x_0,x_1,\dots,x_n\}, \quad a=x_0 < x_1 < \cdots < x_n=b </math> be a partition of the interval..."

New page

The '''Darboux integral''' is a formulation of integration in [[real analysis]] defined using upper and lower sums over partitions of an interval. It provides an order-theoretic approach to integration and is equivalent to the [[Riemann integral]].

== Definition ==

=== Darboux sums ===
Let <math>f:[a,b]\to\mathbb{R}</math> be a bounded function. Let <math display="block">
P=\{x_0,x_1,\dots,x_n\}, \quad a=x_0 < x_1 < \cdots < x_n=b
</math> be a partition of the interval <math>[a,b]</math>.

For each subinterval <math>[x_{i-1},x_i]</math> of <math>P</math>, define:

* the infimum:

<math display="block">
m_i = \inf_{x\in[x_{i-1},x_i]}f(x),
</math>

* the supremum:

<math display="block">
M_i = \sup_{x\in[x_{i-1},x_i]}f(x).
</math>

The '''lower Darboux sum''' of <math>f</math> with respect to <math>P</math> is
<math display="block">
L(f,P)=\sum_{i=1}^n m_i(x_{i}-x_{i-1}),
</math>

and the '''upper Darboux sum''' is
<math display="block">
U(f,P)=\sum_{i=1}^n M_i(x_{i}-x_{i-1}).
</math>

=== The Darboux integral ===
Let <math>\mathcal{P}([a,b])</math> denote the set of all partitions of <math>[a,b]</math>.

The '''lower Darboux integral''' of <math>f</math> on <math>[a,b]</math> is defined by <math display="block">\underline{\int_a^b}f=\sup_{P\in\mathcal{P}([a,b])}L(f,P).</math>

and the '''upper Darboux integral''' is defined by <math display="block">\overline{\int_a^b}f=\inf_{P\in\mathcal{P}([a,b])}U(f,P).</math>

If the upper and lower Darboux integrals are equal, then the '''Darboux integral''' of <math>f</math> on <math>[a,b]</math> is defined by their common value, that is,
<math display="block">\int_a^b f=\underline{\int_a^b}f=\overline{\int_a^b}f.</math>
In this case, function <math>f</math> is said to be '''Darboux-integrable'''.

← Older revision		Revision as of 19:09, 10 April 2026
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	{{Terminology_table\|		{{Terminology_table\|
	{{Terminology_table/row \| Darboux integral \| intégrale de Darboux \| Darboux-Integral \| Darboux 积分 \| Darboux 積分 \| ダルブー積分 }}		{{Terminology_table/row \| Darboux integral \| intégrale de Darboux \| Darboux-Integral \| Darboux 积分 \| Darboux 積分 \| ダルブー積分 }}
			{{Terminology_table/row \| partition \| partition \| Partition \| 分割 \| 分割 \| 分割 }}
	{{Terminology_table/row \| Darboux sum \| somme de Darboux \| Darboux-Summe \| Darboux 和 \| Darboux 和 \| ダルブー和 }}		{{Terminology_table/row \| Darboux sum \| somme de Darboux \| Darboux-Summe \| Darboux 和 \| Darboux 和 \| ダルブー和 }}
	{{Terminology_table/row \| upper Darboux sum \| somme supérieure de Darboux \| obere Darboux-Summe \| 上 Darboux 和 \| 上 Darboux 和 \| 上ダルブー和 }}		{{Terminology_table/row \| upper Darboux sum \| somme supérieure de Darboux \| obere Darboux-Summe \| 上 Darboux 和 \| 上 Darboux 和 \| 上ダルブー和 }}