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The following question from Furdui's book (Exercise 1.32. page 6) is an "open problem" :

Let $f: [0,1] \to \mathbb{R}$ be a continuous (and not a continuously differentiable) function and let

$$x_n = f\left(\dfrac{1}{n}\right) + f\left(\dfrac{2}{n}\right) + \dots + f\left(\dfrac{n-1}{n}\right).$$

Calculate $\lim_{n \to \infty} (x_{n+1}-x_n).$

I am very interested in doing research on this problem but I am not sure whether the problem still is unsolved. [The book has written in 2010(?)] I searched the internet but I couldn't find anything about it esp. searching a long formula without a name attached to it is more difficult (to me). I am new in research and there are lots of difficulties to me (e.g. lack of access to an adviser) to find out proper clear answers to the following questions:

1- Is the mentioned problem unsolved, yet?

2- How to find out all the signs of progress have been done on the specific problem? (It is much easier to gather most of the related things about, say, Catalan's constant but I have no idea about ways of searching an-exercise-of-a-book looking problem in ArXiv or printed journals).

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Emma
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  • The answer works when $x_n/n$ approaches $\int_0^1, f(x),dx$ fast enough, but I should either show that this always happens or else that there are counter examples. – Fimpellizzeri Sep 11 '18 at 21:35
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    @Fimpellizieri, (as suggested by the book) for differentiable function application of Lagrange's mean value theorem works. But I am trying to find a counter-example or why differentiability is necessary ... – Emma Sep 11 '18 at 21:45
  • What a nice question... Trying to work with Hölder functions might be a start. – Did Sep 12 '18 at 18:58
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    I would start by tracking down the source of the problem. See if Furdui leaves bibliographic note somewhere that gives this info. If not, reach out to Furdui. Once you have the source, you can use Google Scholar to see works that cite the source. That should answer both your questions. – yurnero Sep 12 '18 at 19:51
  • By Stolz-Cesaro since $x_n/n\to \int_0^1,f(x),dx$, it suffices to show that $\lim_{n\to\infty} x_{n+1}-x_n$ exists. – Fimpellizzeri Sep 12 '18 at 20:20
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    This problem has been answered some time ago here: https://math.stackexchange.com/questions/569750/speed-of-convergence-of-riemann-sums. In this answer it is also shown that in general the limit $\lim_{n\to\infty}(x_{n+1}-x_n)$ does NOT exist for arbitrary continuous $f$. – sranthrop Sep 12 '18 at 20:50
  • @sranthrop, Thanks a lot :) . Before posting the question, I searched the MSE for a while and I couldn't find the page. Could you let me know how you found it? I am trying to find materials by inputting equations but that's not that easy or even possible sometimes... – Emma Sep 12 '18 at 22:07
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    Uhm... actually it was an accident :) I was looking for estimates on how fast Riemann sums can converge (using google) – sranthrop Sep 13 '18 at 01:21

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