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$$ f(x) = \begin{cases} x^2, & 0 ≤ x ≤ 1 \\[2ex] x+1, & 1 < x ≤ 2 \end{cases} $$

Let $ε > 0$ be given

Question

Let $n ∈ \mathbb N$ be given. Explain why there is a Partition $P = \{\mathbf{X}_0,\mathbf{X}_1,\mathbf{X}_2,\ldots,\mathbf{X}_k\}$ of the interval $[0, 1- 1/n]$ for which $\sum_{i=1}^k (M_i-m_i)(X_i - X_{i-1}) < 1/3ε$

how do i even start with this ? Sorry for reposting I made a error in the previous post so the question did not make sense

I think the following theorem may be of use:

Let $f$ be defined and bounded on $[a,b]$. Then $f$ is Riemann integrable on $[a,b]$ iff for every $ε > 0$ there exists a partition $P$ of $[a,b]$ such that $U(P) - L(P) < ε$

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closed as unclear what you're asking by Did, Shailesh, Daniel W. Farlow, hardmath, user223391 Nov 5 '16 at 7:15

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Essentially you have to explain that if you choose a number $k$ of points $x_0 , x_1 , ... , x_k$ in the interval $[0,1-1/n]$, $\sum_{i=0}^k (x_{i}^2 - x_{i-1}^2 )(x_i - x_{i-1}) < \epsilon$; do you see why? (See also the answer below). $\endgroup$ – Pythagoricus Nov 3 '16 at 9:38
  • $\begingroup$ "how do i even start with this ?" You could start by applying the very specific suggestions made on the other page... or the "game" can continue forever. $\endgroup$ – Did Nov 3 '16 at 9:38
  • $\begingroup$ @Pythagoricus Not true for every partition of size $k$. $\endgroup$ – Did Nov 3 '16 at 9:39
  • $\begingroup$ @Did I didn't mean that; $k$ is any number we might choose to do the work (lost in translation...) $\endgroup$ – Pythagoricus Nov 3 '16 at 9:43
  • $\begingroup$ @Pythagoricus Again: choosing $k$ large enough guarantees nothing. $\endgroup$ – Did Nov 3 '16 at 9:44
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For fixed $n \in \mathbb{N}$, the function $f$ coincides with $x \mapsto x^2$ on the interval $[0,1-1/n]$. If you know that a continuous function is Riemann integrable on a closed interval, you are done. Otherwise, you could also try to mimick the proof that an increasing function is integrable, as you can read in Chapter 6 of Rudin's Principles of Mathematical Analysis.

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  • $\begingroup$ Thank you so much :) They gave us a Theorem that saying if f is monotone then it is riemann integrable $\endgroup$ – Anrich Nov 3 '16 at 10:01

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