Question

Suppose that $h:[0,1]\rightarrow \mathbb{R}$ is a bounded function with following Lower and Upper Integrals:

$\int _{\underline0}^{1}h = 0$ and   $\int _{0}^{\overline1}h = 1$

a) Prove that for every $\epsilon > 0$ , there exists a partition $\alpha$ of such that the difference between the upper and lower sums satisfies $1 \leq U(\alpha,h) - L(\aplha,h) < 1 + \epsilon$ .

b) Furthermore, does there have to be a subdivision such that $U(\alpha,h) - L(\aplha,h) = 1$ . Either prove it or find a counterexample and show to the contrary.

Answer 1

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