Consider one of the standard methods used for defining the Riemann integrals:
Suppose $\sigma$ denotes any subdivision $a=x_0
, 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?