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Give an example of a continuous real-valued function $f$ form $[0,1]$ to $[0,1]$ which takes on every value in $[0,1]$ an infinite number of times.

My example goes like this: Take $f(x)=\lim_{n \to \infty} g(x)$ where. $$g(x)=\left\{\begin{array}{l} n x, \frac{k}{n} \leq x \leq \frac{k+1}{n} ; k \text { is even } \\ -n x, \frac{k}{n} \leq x \leq \frac{k+1}{n} ; k \text { is odd } \end{array}\right.$$ Does this work? I'm unable to verify.

zaemon_23
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    Use MathJax to format your questions. – 5xum Nov 07 '23 at 10:33
  • Can downvoter please explain. Thank you. – zaemon_23 Nov 07 '23 at 10:33
  • @5xum that comment was made before I saw your comment. Also thank you, I did indeed tried Latex before, but for some reason it was showing error, you may check the latex in edit history. That's why I attached pic instead. – zaemon_23 Nov 07 '23 at 10:35
  • Here's the latex that it's showing error to $$g(x)=\left{\begin{array}{l} n x, \frac{k}{n} \leq x \leq \frac{k+1}{n} ; k \text { is even } \ -n x, \frac{k}{n} \leq x \leq \frac{k+1}{n} ; k \text { is odd } \end{array}\right$$ – zaemon_23 Nov 07 '23 at 10:37
  • You should use begin{cases} [insert thing] \end{cases} instead of trying to use left and right with no delimiter behind right. – Bruno B Nov 07 '23 at 10:41
  • Try using the cases environment instead of array. If you simply followed the link I provided you would easily find this: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference/5025#5025 – 5xum Nov 07 '23 at 10:41
  • Are you sure the question is for a function from $[0,1]$ to $[0,1]$, not from $(0,1)$ to $[0,1]$? – 5xum Nov 07 '23 at 10:48
  • @5xum yes that's how it was given. It's from 1959 Putnam B3. For $(0,1)$, $\sin (1/x)$ would work fine – zaemon_23 Nov 07 '23 at 10:50
  • @zaemon_23 Then I think I have a pretty solid proof that $f$ cannot exist. I could be wrong but I can't really see where (which... you know... is always the case until one gets proven wrong, of course :D) – 5xum Nov 07 '23 at 11:07
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    This is quite a hard question. There are examples here and here. – Izaak van Dongen Nov 07 '23 at 14:04
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    Like I said, I could be wrong :D – 5xum Nov 07 '23 at 14:15
  • Is the question really "give an example" (in the sense of explicitly writing down such a function) or "prove such a function exists"? – Andrew D. Hwang Dec 08 '23 at 17:05

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