Darboux Sum

Definition

1. Let $a,b\in R,a<b$
2. Suppose $f$ is Bounded on $[a,b]$ ($\exists c\in R,\forall x\in [a,b],|f(x)|<c$)
3. The lower Darboux sum = $L(f,p)={\sum }_{k=1}^n{m}_i(x_i-x_{i-1})$
   1. Where, $m_i$ is $inf\{f(x)|x\in \{x_{i-1}\dots x_i\}\}$ Infimum
4. The upper Darboux sum = $U(f,p)={\sum }_{k=1}^n{M}_i(x_i-x_{i+1})$
   1. Where, $M_i$ is $sup\{f(x)|x\in \{x_{i-1}\dots x_i\}\}$ Supremum

Darboux Sum

Lower Darboux Sum ($L(f,p)$)

The heights of each rectangle will be the $m_i$

Upper Darboux Sum ($U(f,p)$)

The heights of each rectangle will be $M_i$