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Suppose $f : [a, b] \to \R$, and $\mathcal{P}, \mathcal{Q}$ are partitions of $[a, b]$.
Then $L(f, \mathcal P) \leq U(f, \mathcal Q)$...
- ...always.
- ...provided either $\mathcal{P}$ refines $\mathcal{Q}$ or $\mathcal{Q}$ refines $\mathcal{P}$.
- ...provided $\mathcal{P} = \mathcal{Q}$.
- ...provided $f$ is integrable.
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Suppose $f, g : [a, b] \to \R$ are differentiable on $(a, b)$, that $f(a) \leq g(a)$, and that $f'(t) \leq g'(t)$ for all $t \in (a, b)$.
Then $f(b) \leq g(b)$...
- ...always
- ...provided $f$ is continuous at $a$ and $b$.
- ...provided that $f$ and $g$ are continuous at $a$ and $b$.
- ...provided $f$ and $g$ are continuous at $a$ and $b$, and $f'(t) = g'(t)$ at only finitely many $t \in (a, b)$.
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Suppose that $f : [a, b] \to \R$.
Which of the following is true, and could concievably be useful in a proof?
- For any $\epsilon \gt 0$ there is a partition $\mathcal{P}$ of $[a, b]$ so that $U(f, \mathcal{P}) \lt U(f) - \epsilon$.
- For any $\epsilon \gt 0$ there is a partition $\mathcal{P}$ of $[a, b]$ so that $U(f, \mathcal{P}) \lt U(f) + \epsilon$.
- For any $\epsilon \gt 0$ there is a partition $\mathcal{P}$ of $[a, b]$ so that $U(f, \mathcal{P}) \gt U(f) - \epsilon$.
- For any $\epsilon \gt 0$ there is a partition $\mathcal{P}$ of $[a, b]$ so that $U(f, \mathcal{P}) \gt U(f) + \epsilon$.
(Also, consider: of the other answers, which are false and which are true but useless?)
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Suppose that $f, g : [a, b] \to \R$ and we wish to prove that $U(f+g) \leq U(f) + U(g)$.
Which inequality is true and will be useful in the proof?
- For any set non-empty $I \subseteq[a, b]$, $\sup_{x \in I}(f+g)(x) \leq \sup_{x\in I}f(x) + \sup_{y\in I}g(y)$.
- For any set non-empty $I \subseteq[a, b]$, $\sup_{x \in I}(f+g)(x) \geq \sup_{x\in I}f(x) + \sup_{y\in I}g(y)$.
- For any partition $\mathcal{P}$, we have $U(f, \mathcal{P}) \leq U(f+g, \mathcal{P})$.
- For any partition $\mathcal{P}$, there is a partition $\mathcal{Q}$ refining $\mathcal{P}$ so that $U(f + g, \mathcal{P}) \leq \frac12U(f, \mathcal{Q})$.
