What's the "limit" in the definition of Riemann integrals? Consider one of the standard methods used for defining the Riemann integrals:  

Suppose $\sigma$ denotes any subdivision $a=x_0<x_1<x_2\cdots<x_{n-1}<x_n=b$, and let $x_{i-1}\leq \xi_i\leq x_i$. Then if 
  $$|\sigma|:=\max\{x_i-x_{i-1}|i=1,\cdots,n\},$$
  which we shall call the norm of the subdivision, we define:
  $$\int_a^bf(x)dx:=\lim_{|\sigma|\to 0}\sum_{i=1}^nf(\xi_i)(x_i-x_{i-1}).$$

When one talks about the limit of a function $\lim_{x\to x_0}f(x)$, one has exactly one value $f(x)$ for every $x$. However, for every $|\sigma|$, the value of the Riemann sum $\sum_{i=1}^nf(\xi_i)(x_i-x_{i-1})$ is not necessarily unique. Using the $\epsilon$-$\delta$ language, one may restate the definition as follows:

Suppose $f:[a,b]\to{\mathbb R}$, $J\in{\mathbb R}$. If for all $\epsilon>0$, there exists $\delta>0$ such that for any subdivision $\sigma$ and $\{\xi_i\}$ on $\sigma$ (i.e. $x_{i-1}\leq \xi_i\leq x_i$), $|\sigma|<\delta$ implies
  $$|\sum_{i=1}^nf(\xi_i)\Delta x_i-J|<\epsilon,$$
  we call $J$ is the Riemann integral of $f$ on $[a,b]$ and denote
  $$J=\int_a^bf(x)dx.$$

Here are my questions:


*

*How should I understand this kind of limit?

*It seems that this is not the "limit of a function" I learned in elementary real analysis. Where does it appear in mathematics besides the definition of Riemann integrals?

 A: It can be stated in terms of the ordinary definition of limit.  Let $A(\sigma)$ and $B(\sigma)$ respecively be the supremum and infimum of $\sum_i f(\xi_i) (x_i - x_{i-1})$ over all subdivisions of "norm" $\sigma$ and all choices of the $\xi_i$.  Then if 
$\lim_{\sigma \to 0} A(\sigma) = \lim_{\sigma \to 0} B(\sigma)$, i.e. both limits exist and are equal, the common value is the Riemann integral.
A: It is the limit of a net. Nets are a generalization of sequences which make all the familiar statements about sequences true for spaces that are not first-countable (for example a point lies in the closure of a subspace if and only if there is a net converging to it, and so forth), so any time you want to prove something about general spaces and you would like to use sequences but can't, you can use nets instead (although there are some subtleties here; one cannot just replace "sequence" with "net" in a proof). 
A: Qiaochu Yuan and minimalrho explained very well how to use nets.
Filters (or filter bases) also can be used to formalize the concept of Riemann integral. Nets and filters are important tools in topology and functional analysis.
Just for completeness' sake, I would like to mention here another generalization of limit: G tends to b as F tends to a.
Let $S$ be a set, $X,Y$ be topological spaces,
$F\colon S\to X$, $G\colon S\to Y$, $a\in X$, $b\in Y$.
We say that $G\to b$ as $F\to a$ if for every neighborhood $V$ of $b$
there exists a neighborhood $U$ of $a$ such that for every $s\in S$
the condition $F(s)\in U$ implies that $G(s)\in V$.
This concept of limit is not so powerful as nets and filters
(and it can be reduced to nets or filters), but it is very close to the definition of Riemann integral. In the definition of Riemann integral, $S$ is the set of tagged partitions, $F$ is the norm of the partition and $G$ is the integral sum.
A: One way of thinking about it is that you have a function defined on the set of partitions of $[a,b]$ into the real numbers called the Riemann sum. You put an order on partitions by defining the notion of mesh ($|\sigma|$ in your notation) and defining an order on the set of partitions by $\sigma\succeq\tau$, if and only if $|\sigma| \leq |\tau|$ and say that $\sigma$ is finer than $\tau$. So now you can make a definition similar to the limit of sequences: $\lim_{|\sigma|\rightarrow 0} R(\sigma)=J$ if and only if for all $\epsilon>0$ there exists a partition $\Lambda$ such that for all partitions $\sigma$ such that $\sigma\succeq\Lambda$ one has $|R(\sigma)-J|<\epsilon$.
The more general context for this is that we are making the set of partitions into a directed set, and so Riemann sum becomes a net from the set of partitions into $\mathbb{R}$.
A: Besides taking the limit of a function, you can take the limit of any relation, thought of as a multi-valued function.  Recall that $ \lim _ { x \to x _ 0 } y = L $, where $ y = f ( x ) $ for some function $ f $, means that there is a unique $ L $ such that, for each $ \epsilon > 0 $, for some $ \delta > 0 $, for each $ x $ in the domain of $ f $, if $ 0 < \lvert x - x _ 0 \rvert < \delta $ and $ y = f ( x ) $, then $ | y - L | < \epsilon $.  Compared to what you'll usually see in Calculus textbooks, I wrote $ y $ instead of $ f ( x ) $, so that I had to throw in the statement that $ y = f ( x ) $, but this is obviously equivalent.  To avoid dummy variables, we can also say that $ L $ is the limit of $ f $ approaching $ x _ 0 $.
Similarly, $ \lim _ { x \to x _ 0 } y = L $, where now $ R ( x , y ) $ for some relation $ R $, means that there is a unique real number $ L $ such that, for each $ \epsilon > 0 $, for some $ \delta > 0 $, for each $ x $ in the domain of $ R $, if $ 0 < \lvert x - x _ 0 \rvert < \delta $ and $ R ( x , y ) $, then $ | y - L | < \epsilon $.  Compared to the previous statement, I replaced $ y = f ( x ) $ with $ R ( x , y ) $, and otherwise this is identical.  To avoid dummy variables again, we now say that $ L $ is the limit of $ R $ approaching $ x _ 0 $.  Of course, if $ R $ is a functional (single-valued) relation, then this agrees with the previous paragraph.
For a simple example, let $ R ( x , y ) $ mean that $ \lvert x \rvert = \lvert y \rvert $, or $ y = \pm x $ to look like a multi-valued function.  Note that every real number is in the domain of $ R $.  Then $ \lim _ { x \to 0 } ( \pm x ) = 0 $, because for each $ \epsilon > 0 $, for some $ \delta > 0 $ (such as $ \delta = \epsilon $ in this case), for each real number $ x $, if $ 0 < \lvert x - 0 \rvert < \delta $ and $ y = \pm x $, then $ \lvert y - 0 \rvert < \epsilon $.
For the Riemann integral, $ R ( x , y ) $ means that $ y $ is the value of a Riemann sum whose partition has norm $ x $; it might be good to use different variables names here, so I'll use $ n $ (for norm) and $ S $ (for sum) instead.  The domain of this relation (the set of possible norms) is the set of positive numbers.  So, to say that $ L $ is the limit of $ R $ approaching $ 0 $ is to say that, for each $ \epsilon > 0 $, for some $ \delta > 0 $, for each positive number $ N $, if $ 0 < \lvert N - 0 \rvert < \delta $ and $ I $ is the value of a Riemann sum whose partition has norm $ N $, then $ | I - L | < \epsilon $.  Since $ N $ is positive, we can simplify $ 0 < \lvert N - 0 \rvert < \delta $ to $ N < \delta $, and now we have the usual definition of the Riemann integral.
