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In [[real analysis]], a branch of [[mathematics]], the '''Darboux integral''' is constructed using '''Darboux sums''' and is one possible definition of the [[integral]] of a [[function (mathematics)|function]]. Darboux integrals are equivalent to [[Riemann integral]]s, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal.<ref>{{cite book|author1=David J. Foulis|author2=Mustafa A. Munem|title=After Calculus: Analysis|url=https://books.google.com/books?id=kSMnAQAAIAAJ|year=1989|publisher=Dellen Publishing Company|isbn=978-0-02-339130-9|page=396}}</ref> The definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral. Consequently, introductory textbooks on [[calculus]] and real analysis often develop Riemann integration using the Darboux integral, rather than the true Riemann integral.<ref>{{Cite book|title=Calculus (3rd. edition)|url=https://archive.org/details/calculus00spiv_191|url-access=limited|last=Spivak|first=M.|publisher=Publish Or Perish, Inc.|year=1994|isbn=0-914098-89-6|location=Houston, TX|pages=[https://archive.org/details/calculus00spiv_191/page/n266 253]–255}}</ref> Moreover, the definition is readily extended to defining [[Riemann–Stieltjes integration]].<ref>{{Cite book|title=Principles of Mathematical Analysis (3rd. edition)|url=https://archive.org/details/principlesmathem00rudi_663|url-access=limited|last=Rudin|first=W.|publisher=McGraw-Hill|year=1976|isbn=007054235X|location=New York|pages=[https://archive.org/details/principlesmathem00rudi_663/page/n128 120]–122}}</ref> Darboux integrals are named after their inventor, [[Gaston Darboux]].

==Definition==
The definition of the Darboux integral considers '''upper and lower (Darboux) integrals''', which exist for any [[bounded function|bounded]] [[real number|real]]-valued function <math>f</math> on the [[interval (mathematics)|interval]] <math>[a,b]</math>. The '''Darboux integral''' exists if and only if the upper and lower integrals are equal. The upper and lower integrals are in turn the [[infimum and supremum]], respectively, of '''upper and lower (Darboux) sums''' which over- and underestimate, respectively, the "area under the curve." In particular, for a given partition of the interval of integration, the upper and lower sums add together the areas of rectangular slices whose heights are the supremum and infimum, respectively, of ''f'' in each subinterval of the partition. These ideas are made precise below:

===Darboux sums===
A [[partition of an interval]] <math>[a,b]</math> is a finite sequence of values ''x''''i'' such that

:<math>a = x_0 < x_1 < \cdots < x_n = b. </math>

Each interval [''x''''i''−1, ''x''''i''] is called a ''subinterval'' of the partition. Let ''f'': [''a'', ''b''] → '''R''' be a bounded function, and let

:<math>P = (x_0, \ldots, x_n)</math>

be a partition of [''a'', ''b'']. Let

:<math>\begin{align}
M_i = \sup_{x\in[x_{i-1},x_{i}]} f(x) , \\
m_i = \inf_{x\in[x_{i-1},x_{i}]} f(x) .
\end{align}</math>

[[Image:Darboux.svg|thumb|right|Lower (green) and upper (green plus lavender) Darboux sums for four subintervals]]

The '''upper Darboux sum''' of ''f'' with respect to ''P'' is

:<math>U_{f, P} = \sum_{i=1}^n (x_{i}-x_{i-1}) M_i . \,\!</math>

The '''lower Darboux sum''' of ''f'' with respect to ''P'' is

:<math>L_{f, P} = \sum_{i=1}^n (x_{i}-x_{i-1}) m_i . \,\!</math>

The lower and upper Darboux sums are often called the lower and upper sums.

===Darboux integrals===
The '''upper Darboux integral''' of ''f'' is

:<math>U_f = \inf\{U_{f,P} \colon P \text{ is a partition of } [a,b]\} . \,\!</math>

The '''lower Darboux integral''' of ''f'' is

:<math>L_f = \sup\{L_{f,P} \colon P \text{ is a partition of } [a,b]\} . \,\!</math>

In some literature an integral symbol with an underline and overline represent the lower and upper Darboux integrals respectively.

:<math>\begin{align}
L_f \equiv \underline{\int_{a}^{b}} f(x) \, \mathrm{d}x \\[6pt]
U_f \equiv \overline{\int_{a}^{b}} f(x) \, \mathrm{d}x,
\end{align}</math>

and like Darboux sums they are sometimes simply called the lower and upper integrals.

If ''U''''f'' = ''L''''f'', then we call the common value the '''Darboux integral'''.<ref>Wolfram MathWorld</ref> We also say that ''f'' is ''Darboux-integrable'' or simply ''integrable'' and set

:<math>\int_a^b {f(t)\,dt} = U_f = L_f ,</math>

An equivalent and sometimes useful criterion for the integrability of ''f'' is to show that for every ε > 0 there exists a partition ''P''ε of [''a'', ''b''] such that<ref>Spivak 2008, chapter 13.</ref>

:<math> U_{f,P_\epsilon} - L_{f,P_\epsilon} < \varepsilon.</math>

==Properties==
*For any given partition, the upper Darboux sum is always greater than or equal to the lower Darboux sum. Furthermore, the lower Darboux sum is bounded below by the rectangle of width (''b''−''a'') and height inf(''f'') taken over [''a'', ''b'']. Likewise, the upper sum is bounded above by the rectangle of width (''b''−''a'') and height sup(''f'').
*:<math>(b-a)\inf_{x \in [a,b]} f(x) \leq L_{f,P} \leq U_{f,P} \leq (b-a)\sup_{x \in [a,b]} f(x)</math>
*The lower and upper Darboux integrals satisfy
*:<math>\underline{\int_{a}^{b}} f(x) \, dx \leq \overline{\int_{a}^{b}} f(x) \, dx</math>
*Given any ''c'' in (''a'', ''b'')
*:<math>\begin{align}
\underline{\int_{a}^{b}} f(x) \, dx &= \underline{\int_{a}^{c}} f(x) \, dx + \underline{\int_{c}^{b}} f(x) \, dx\\[6pt]
\overline{\int_{a}^{b}} f(x) \, dx &= \overline{\int_{a}^{c}} f(x) \, dx + \overline{\int_{c}^{b}} f(x) \, dx
\end{align}</math>
*The lower and upper Darboux integrals are not necessarily linear. Suppose that ''g'':[''a'', ''b''] → '''R''' is also a bounded function, then the upper and lower integrals satisfy the following inequalities.
*:<math>\begin{align}
\underline{\int_{a}^{b}} f(x) \, dx + \underline{\int_{a}^{b}} g(x) \, dx &\leq \underline{\int_{a}^{b}} (f(x) + g(x)) \, dx\\[6pt]
\overline{\int_{a}^{b}} f(x) \, dx + \overline{\int_{a}^{b}} g(x) \, dx &\geq \overline{\int_{a}^{b}} (f(x) + g(x)) \, dx
\end{align}</math>
*For a constant ''c'' ≥ 0 we have
*:<math>\begin{align}
\underline{\int_{a}^{b}} cf(x) &= c\underline{\int_{a}^{b}} f(x)\\[6pt]
\overline{\int_{a}^{b}} cf(x) &= c\overline{\int_{a}^{b}} f(x)
\end{align}</math>
*For a constant ''c'' ≤ 0 we have
*:<math>\begin{align}
\underline{\int_{a}^{b}} cf(x) &= c\overline{\int_{a}^{b}} f(x)\\[6pt]
\overline{\int_{a}^{b}} cf(x) &= c\underline{\int_{a}^{b}} f(x)
\end{align}</math>
*Consider the function: <math display="block">\begin{align}
&{} F : [a, b] \to \R \\
&{} F(x) = \underline{\int_{a}^{x}} f(t) \, dt
\end{align}</math> then ''F'' is [[Lipschitz continuous]]. An identical result holds if ''F'' is defined using an upper Darboux integral.

==Examples==

===A Darboux-integrable function===
Suppose we want to show that the function ''f''(''x'') = ''x'' is Darboux-integrable on the interval [0, 1] and determine its value. To do this we partition [0, 1] into ''n'' equally sized subintervals each of length 1/''n''. We denote a partition of ''n'' equally sized subintervals as ''P''''n''.

Now since ''f''(''x'') = ''x'' is strictly increasing on [0, 1], the infimum on any particular subinterval is given by its starting point. Likewise the supremum on any particular subinterval is given by its end point. The starting point of the ''k''th subinterval in ''P''''n'' is (''k''−1)/''n'' and the end point is ''k''/''n''. Thus the lower Darboux sum on a partition ''P''''n'' is given by

:<math>\begin{align}
L_{f,P_n} &= \sum_{k = 1}^{n} f(x_{k-1})(x_{k} - x_{k-1}) \\
&= \sum_{k = 1}^{n} \frac{k-1}{n} \cdot \frac{1}{n} \\
&= \frac{1}{n^2} \sum_{k = 1}^{n} [k-1] \\
&= \frac{1}{n^2}\left[ \frac{(n-1)n}{2} \right]
\end{align}</math>

similarly, the upper Darboux sum is given by

:<math>\begin{align}
U_{f,P_n} &= \sum_{k = 1}^{n} f(x_{k})(x_{k} - x_{k-1}) \\
&= \sum_{k = 1}^{n} \frac{k}{n} \cdot \frac{1}{n} \\
&= \frac{1}{n^2} \sum_{k = 1}^{n} k \\
&= \frac{1}{n^2}\left[ \frac{(n+1)n}{2} \right]
\end{align}</math>

Since
:<math>U_{f,P_n} - L_{f,P_n} = \frac{1}{n}</math>
Thus for given any ε > 0, we have that any partition ''P''''n'' with <math>n > \frac{1}{\epsilon}</math> satisfies
:<math>U_{f,P_n} - L_{f,P_n} < \epsilon</math>
which shows that ''f'' is Darboux integrable. To find the value of the integral note that

:<math>\int_{0}^{1}f(x) \, dx
= \lim_{n \to \infty} U_{f,P_n}
= \lim_{n \to \infty} L_{f,P_n}
= \frac{1}{2}</math>
{{multiple image

| align = center
| direction = horizontal
| width = 300

| header = Darboux sums
| header_align = center
| header_background =
| footer =
| footer_align =
| footer_background =
| background color =

|image1=Riemann Integration and Darboux Upper Sums.gif
|width1=300
|caption1=<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">Darboux upper sums of the function {{math|1=''y'' = ''x''2}}</div>
|alt1=Upper Darboux sum example

|image2=Riemann Integration and Darboux Lower Sums.gif
|width2=300
|caption2=<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">Darboux lower sums of the function {{math|1=''y'' = ''x''2}}</div>
|alt2=Lower Darboux sum example
}}

===An unintegrable function===
Suppose we have the function <math>f:[0,1] \to \R</math> defined as

:<math>\begin{align}
f(x) &=
\begin{cases}
0 & \text{if }x\text{ is rational} \\
1 & \text{if }x\text{ is irrational}
\end{cases}
\end{align}</math>

Since the rational and irrational numbers are both [[dense subset]]s of '''R''', it follows that ''f'' takes on the value of 0 and 1 on every subinterval of any partition. Thus for any partition ''P'' we have

:<math>\begin{align}
L_{f,P} &=\sum_{k = 1}^{n}(x_{k} - x_{k-1})\inf_{x \in [x_{k-1},x_{k}]}f = 0 \\
U_{f,P} &=\sum_{k = 1}^{n}(x_{k} - x_{k-1}) \sup_{x \in [x_{k-1},x_{k}]}f = 1
\end{align}</math>
from which we can see that the lower and upper Darboux integrals are unequal.

==Refinement of a partition and relation to Riemann integration==
[[Image:Darboux refinement.svg|250px|thumb|right|When passing to a refinement, the lower sum increases and the upper sum decreases.]]
A ''refinement'' of the partition <math>x_0, \ldots, x_n</math> is a partition <math>y_0, \ldots, y_m</math> such that for all ''i'' = 0, …, ''n'' there is an [[integer]] ''r''(''i'') such that

:<math> x_{i} = y_{r(i)} . </math>

In other words, to make a refinement, cut the subintervals into smaller pieces and do not remove any existing cuts.

If <math>P' = (y_0,\ldots,y_m) </math> is a refinement of <math>P = (x_0,\ldots,x_n) , </math> then

:<math>U_{f, P} \ge U_{f, P'} </math>

and

:<math>L_{f, P} \le L_{f, P'}. </math>

If ''P''1, ''P''2 are two partitions of the same interval (one need not be a refinement of the other), then

:<math>L_{f, P_1} \le U_{f, P_2}, </math>

and it follows that

:<math>L_f \le U_f . </math>

Riemann sums always lie between the corresponding lower and upper Darboux sums. Formally, if <math>P = (x_0,\ldots,x_n) </math> and <math>T = (t_1,\ldots,t_n) </math> together make a tagged partition

:<math> x_0 \le t_1 \le x_1\le \cdots \le x_{n-1} \le t_n \le x_n </math>

(as in the definition of the [[Riemann integral]]), and if the Riemann sum of <math>f</math> corresponding to ''P'' and ''T'' is ''R'', then

:<math>L_{f, P} \le R \le U_{f, P}. </math>

From the previous fact, Riemann integrals are at least as strong as Darboux integrals: if the Darboux integral exists, then the upper and lower Darboux sums corresponding to a sufficiently fine partition will be close to the value of the integral, so any Riemann sum over the same partition will also be close to the value of the integral. There is (see below) a tagged partition that comes arbitrarily close to the value of the upper Darboux integral or lower Darboux integral, and consequently, if the Riemann integral exists, then the Darboux integral must exist as well.

:{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Details of finding the tags
|-
|
For this proof, we shall use superscripts to index <math>\left\{ P^{(n)} \right\}</math> and variables related to it.

Let <math>\left\{ P^{(n)} \right\}</math> be an arbitrary partition of <math>[a, b]</math> such that <math>\|P_n\|\to 0</math>, whose tags are to be determined.

By the definition of infimum, for
any <math>\epsilon > 0</math>, we can always find a <math>t^{(n)}_i \in
\left[ x^{(n)}_i, x^{(n)}_{i+1} \right] </math>
such that <math>
\inf_{x \in \left[ x^{(n)}_i, x^{(n)}_{i+1} \right] } f(x) \ge
f(t^{(n)}_{i}) - \epsilon. </math>
Thus,
:<math>
\begin{aligned}
\sum_{i=0}^{N^{(n)}-1}
f(t_i)
(x_{i+1}^{(n)} - x_i^{(n)})
& \le &
\sum_{i=0}^{N^{(n)}-1}
\left(
\inf_{x \in \left[ x^{(n)}_i, x^{(n)}_{i+1} \right] } f(x)
+
\epsilon
\right)
(x_{i+1}^{(n)} - x_i^{(n)})
& \ \ \
\\
& = &
\sum_{i=0}^{N^{(n)}-1}
\inf_{x \in \left[ x^{(n)}_i, x^{(n)}_{i+1} \right] } f(x)
(x_{i+1}^{(n)} - x_i^{(n)})
+
\sum_{i=0}^{N-1}
\epsilon
(x_{i+1}^{(n)} - x_i^{(n)})
\\
& = &
\sum_{i=0}^{N^{(n)}-1}
\inf_{x \in \left[ x^{(n)}_i, x^{(n)}_{i+1} \right] } f(x)
(x_{i+1}^{(n)} - x_i^{(n)})
+
\epsilon
(b - a) .
\end{aligned}
</math>
Let <math> \epsilon = 1 / n(b - a) </math>, we have
:<math>
\begin{aligned}
\sum_{i=0}^{N^{(n)}-1}
f(t_i)
(x_{i+1}^{(n)} - x_i^{(n)})
& \le &
\sum_{i=0}^{N^{(n)}-1}
\inf_{x \in \left[ x^{(n)}_i, x^{(n)}_{i+1} \right] } f(x)
(x_{i+1}^{(n)} - x_i^{(n)})
+ \frac{1}{n}
\\
& = &
L_{f, P^{(n)}}
+ \frac{1}{n}
\end{aligned}
</math>
Taking limits of both sides,
:<math>
\begin{aligned}
R_f =
\sum_{i=0}^{N^{(n)}-1}
f(t_i)
(x_{i+1}^{(n)} - x_i^{(n)})
\le
\lim_{n\to \infty}
L_{f, P^{(n)}}
+
\lim_{n\to \infty}
\frac{1}{n}
=
\lim_{n\to \infty}
L_{f, P^{(n)}} .
\end{aligned}
</math>
Similarly, (with a different sequences of tags)
:<math>
\begin{aligned}
R_f
\ge
\lim_{n\to \infty}
U_{f, P^{(n)}} .
\end{aligned}
</math>
Thus, we have
:<math>
R_f
\le
\lim_{n\to \infty} L_{f, P^{(n)}}
\le
\lim_{n\to \infty} U_{f, P^{(n)}}
\le
R_f,
</math>
which means that the Darboux integral exist and equals <math>R_f</math>.

|}

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Darboux_integral Darboux integral, Wikipedia] under a CC BY-SA license

@@ Line 1: / Line 1: @@
-In [[real analysis]], a branch of [[mathematics]], the '''Darboux integral''' is constructed using '''Darboux sums''' and is one possible definition of the [[integral]] of a [[function (mathematics)|function]]. Darboux integrals are equivalent to [[Riemann integral]]s, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal.<ref>{{cite book|author1=David J. Foulis|author2=Mustafa A. Munem|title=After Calculus: Analysis|url=https://books.google.com/books?id=kSMnAQAAIAAJ|year=1989|publisher=Dellen Publishing Company|isbn=978-0-02-339130-9|page=396}}</ref> The definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral. Consequently, introductory textbooks on [[calculus]] and real analysis often develop Riemann integration using the Darboux integral, rather than the true Riemann integral.<ref>{{Cite book|title=Calculus (3rd. edition)|url=https://archive.org/details/calculus00spiv_191|url-access=limited|last=Spivak|first=M.|publisher=Publish Or Perish, Inc.|year=1994|isbn=0-914098-89-6|location=Houston, TX|pages=[https://archive.org/details/calculus00spiv_191/page/n266 253]–255}}</ref>  Moreover, the definition is readily extended to defining [[Riemann–Stieltjes integration]].<ref>{{Cite book|title=Principles of Mathematical Analysis (3rd. edition)|url=https://archive.org/details/principlesmathem00rudi_663|url-access=limited|last=Rudin|first=W.|publisher=McGraw-Hill|year=1976|isbn=007054235X|location=New York|pages=[https://archive.org/details/principlesmathem00rudi_663/page/n128 120]–122}}</ref>  Darboux integrals are named after their inventor, [[Gaston Darboux]].
+In real analysis, a branch of mathematics, the '''Darboux integral''' is constructed using '''Darboux sums''' and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal. The definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral. Consequently, introductory textbooks on calculus and real analysis often develop Riemann integration using the Darboux integral, rather than the true Riemann integral. Moreover, the definition is readily extended to defining Riemann–Stieltjes integration. Darboux integrals are named after their inventor, Gaston Darboux.
 ==Definition==
-The definition of the Darboux integral considers '''upper and lower (Darboux) integrals''', which exist for any [[bounded function|bounded]] [[real number|real]]-valued function <math>f</math> on the [[interval (mathematics)|interval]] <math>[a,b]</math>. The '''Darboux integral''' exists if and only if the upper and lower integrals are equal.  The upper and lower integrals are in turn the [[infimum and supremum]], respectively, of '''upper and lower (Darboux) sums''' which over- and underestimate, respectively, the "area under the curve." In particular, for a given partition of the interval of integration, the upper and lower sums add together the areas of rectangular slices whose heights are the supremum and infimum, respectively, of ''f'' in each subinterval of the partition.  These ideas are made precise below:
+The definition of the Darboux integral considers '''upper and lower (Darboux) integrals''', which exist for any bounded real-valued function <math>f</math> on the interval <math>[a,b]</math>. The '''Darboux integral''' exists if and only if the upper and lower integrals are equal.  The upper and lower integrals are in turn the infimum and supremum, respectively, of '''upper and lower (Darboux) sums''' which over- and underestimate, respectively, the "area under the curve." In particular, for a given partition of the interval of integration, the upper and lower sums add together the areas of rectangular slices whose heights are the supremum and infimum, respectively, of ''f'' in each subinterval of the partition.  These ideas are made precise below:
 ===Darboux sums===
-A [[partition of an interval]] <math>[a,b]</math> is a finite sequence of values ''x''<sub>''i''</sub> such that
+A partition of an interval <math>[a,b]</math> is a finite sequence of values ''x''<sub>''i''</sub> such that
 :<math>a = x_0 < x_1 < \cdots < x_n = b. </math>
@@ Line 50: / Line 50: @@
 and like Darboux sums they are sometimes simply called the lower and upper integrals.
-If ''U''<sub>''f''</sub>&nbsp;=&nbsp;''L''<sub>''f''</sub>, then we call the common value the '''Darboux integral'''.<ref>Wolfram MathWorld</ref> We also say that ''f'' is ''Darboux-integrable'' or simply ''integrable'' and set
+If ''U''<sub>''f''</sub>&nbsp;=&nbsp;''L''<sub>''f''</sub>, then we call the common value the '''Darboux integral'''. We also say that ''f'' is ''Darboux-integrable'' or simply ''integrable'' and set
 :<math>\int_a^b {f(t)\,dt} = U_f = L_f ,</math>
-An equivalent and sometimes useful criterion for the integrability of ''f'' is to show that for every ε > 0 there exists a partition ''P''<sub>ε</sub> of  [''a'',&thinsp;''b''] such that<ref>Spivak 2008, chapter 13.</ref>
+An equivalent and sometimes useful criterion for the integrability of ''f'' is to show that for every ε > 0 there exists a partition ''P''<sub>ε</sub> of  [''a'',&thinsp;''b''] such that
 :<math> U_{f,P_\epsilon} - L_{f,P_\epsilon} < \varepsilon.</math>
@@ Line 86: / Line 86: @@
 &{} F : [a, b] \to \R \\
 &{} F(x) = \underline{\int_{a}^{x}} f(t) \, dt
-\end{align}</math> then ''F'' is [[Lipschitz continuous]]. An identical result holds if ''F'' is defined using an upper Darboux integral.
+\end{align}</math> then ''F'' is Lipschitz continuous. An identical result holds if ''F'' is defined using an upper Darboux integral.
 ==Examples==
@@ Line 121: / Line 121: @@
 = \lim_{n \to \infty} L_{f,P_n}
 = \frac{1}{2}</math>
-|image1=Riemann Integration and Darboux Upper Sums.gif
+<div style="text-align: center;"> Darboux sums </div>
-|width1=300
+<div style="text-align: center;"><gallery widths="400">
-|caption1=<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">Darboux upper sums of the function {{math|1=''y'' = ''x''<sup>2</sup>}}</div>
+File:Riemann Integration and Darboux Upper Sums.gif|Darboux upper sums of the function ''y'' = ''x''<sup>2</sup>
-|alt1=Upper Darboux sum example
+File:Riemann Integration and Darboux Lower Sums.gif|Darboux lower sums of the function ''y'' = ''x''<sup>2</sup>
+</gallery></div>
 ===An unintegrable function===
@@ Line 157: / Line 139: @@
 \end{align}</math>
-Since the rational and irrational numbers are both [[dense subset]]s of '''R''', it follows that ''f'' takes on the value of 0 and 1 on every subinterval of any partition. Thus for any partition ''P'' we have
+Since the rational and irrational numbers are both dense subsets of '''R''', it follows that ''f'' takes on the value of 0 and 1 on every subinterval of any partition. Thus for any partition ''P'' we have
 :<math>\begin{align}
@@ Line 167: / Line 149: @@
 ==Refinement of a partition and relation to Riemann integration==
 [[Image:Darboux refinement.svg|250px|thumb|right|When passing to a refinement, the lower sum increases and the upper sum decreases.]]
-A ''refinement'' of the partition <math>x_0, \ldots, x_n</math> is a partition <math>y_0, \ldots, y_m</math> such that for all ''i'' = 0, …, ''n'' there is an [[integer]] ''r''(''i'') such that
+A ''refinement'' of the partition <math>x_0, \ldots, x_n</math> is a partition <math>y_0, \ldots, y_m</math> such that for all ''i'' = 0, …, ''n'' there is an integer ''r''(''i'') such that
 :<math> x_{i} = y_{r(i)} . </math>
@@ Line 193: / Line 175: @@
 :<math> x_0 \le t_1 \le x_1\le \cdots \le x_{n-1} \le t_n \le x_n </math>
-(as in the definition of the [[Riemann integral]]), and if the Riemann sum of <math>f</math> corresponding to ''P'' and ''T'' is ''R'', then
+(as in the definition of the Riemann integral), and if the Riemann sum of <math>f</math> corresponding to ''P'' and ''T'' is ''R'', then
 :<math>L_{f, P} \le R \le U_{f, P}. </math>