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7.1: Верхній і нижній інтеграли
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Припустимо $x_{i-1}<s<x_{i}$ і нехай \[\begin{aligned} w_{1} &=\inf \left\{f(x): x_{i-1} \leq x \leq s\right\}, \\ W_{1} &=\sup \left\{f(x): x_{i-1} \leq x \leq s\right\}, \\ w_{2} &=\inf \left\{f(x)...Припустимо $x_{i-1}<s<x_{i}$ і нехай $\begin{aligned} w_{1} &=\inf \left\{f(x): x_{i-1} \leq x \leq s\right\}, \\ W_{1} &=\sup \left\{f(x): x_{i-1} \leq x \leq s\right\}, \\ w_{2} &=\inf \left\{f(x): s \leq x \leq x_{i}\right\}, \\ W_{2} &=\sup \left\{f(x): s \leq x \leq x_{i}\right\}, \\ m_{i} &=\inf \left\{f(x): x_{i-1} \leq x \leq x_{i}\right\}, \end{aligned}$ і $M_{i}=\sup \left\{f(x): x_{i-1} \leq x \leq x_{i}\right\}.$ Тоді $w_{1} \geq m_{i}, w_{2} \geq m_{i}, W_{1} \leq M_{i},$ і\(…