Intro

There exists systems modeled by equations of the form $$\tag{1} x(t_{i+1})=x(t_{i})+f(t_{i},x(t_{i}))(t_{i+1}-t_{i}) \ \ \ \ 0=t_{0}<t_{1}<\cdots<t_{N}=1$$“In the limit” as $(t_{0},\dots,t_{N})$ of $[0,1]$ gets finer (i.e. $\max_{i}\mid t_{i+1}-t_{i}\mid\to 0$) leads to an integral equation of the form $$x(t)=x(0)+\int \limits_{0}^tf(s,x(s)) \, ds$$where the integral is the Riemann Integral.

Problems with The Riemann Integral

Let $f_{n}:[0,1]\to \mathbb{R}$ be a sequence of Riemann-integrable functions on $[0,1]$, there are two issues: 1. Assume $\exists f:[0,1]\to \mathbb{R}$ such that $f_{n}(x)\to f(x)$ as $n\to \infty$, $\forall x \in[0,1]$ (i.e. $(f_{n})_{n\in\mathbb{N}}$ converges pointwise on $[0,1]$ to $f$) then, it does not follow that $f$ is Riemann-integrable 2. Even if $f$ happens to be Riemann-integrable, it does not follow that $$\lim_{ n \to \infty } \int\limits_{0}^{1}f_{n}(x) \, dx =\int\limits _{0}^{1}f(x) \, dx$$ i.e., the limit and Riemann integral cannot be always interchanged.

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Definition

A function $f:[0,1]\to \mathbb{R}$ is Riemann integrable if $\forall\epsilon>0, \exists\delta>0$ such that for each subdivision $\pi=(t_{0},\dots,t_{N(\pi)})$ of $[0,1]$ we have that if the subdivision is fine enough then $$\delta(\pi)<\delta \implies|S_{f,\pi}^{u}-S_{f,\pi}^{\mathscr{l}}|<\epsilon$$