A problem from Zorich

Asked May 18 '19 at 16:48

Active May 18 '19 at 17:17

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I am trying to read Mathematical Analysis I by Zorich on my own. Here is an exercise that I could not solve:

For all $l \in \mathbb{R}$ that is not of the form $\frac{1}{n}$ for some $n \in \mathbb{N}$, there exists a continuous function $f: [0, 1] \Rightarrow \mathbb{R}$ such that $f(0) = f(1)$ and the graph of $f$ contains no horizontal chord with length $l$.

I was wondering how to prove this? Thanks in advance.

edited May 18 '19 at 17:17

John Douma

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asked May 18 '19 at 16:48

Ziyang Guo

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Who is Zorich?? Is it the book on Analysis 1? – Dietrich Burde May 18 '19 at 16:49
Yes! It is Mathematical Analysis I by Zorich. – Ziyang Guo May 18 '19 at 16:51
Chapter 4 I think, which is about continuity of functions. – Ziyang Guo May 18 '19 at 16:51
What is a horizontal chord? – Maximilian Janisch May 18 '19 at 16:53
Can you provide the definition of horizontal chord? Is this just a horizontal line connecting any two points of the graph or something else? – Clayton May 18 '19 at 16:53
It is basically saying that there does not exist any x in [0, 1] such that f(x) agrees with f(x + l). – Ziyang Guo May 18 '19 at 16:54
So yes, a horizontal chord is just a horizontal line connecting two points of the graph. – Ziyang Guo May 18 '19 at 16:55
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Please change the title to be more specific (or at least to not include grammar errors)! – ViktorStein May 18 '19 at 16:57

A problem from Zorich

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