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Let, $f:[a,b]$$\rightarrow$$\mathbb{R}$ be a continuous function. Is it possible to find out a partition of $[a,b]$ such that $f$ is monotone there?

I am stuck here. How to proceed from here?

Topology
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1 Answers1

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Not in general. There are continuous functions which are not monotonic on any interval. For example the Weierstrass function.

user71352
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  • Weierstrass function means that function which is cont everywhere but differentiable nowhere? – Topology Jul 03 '14 at 05:58
  • Yes I do mean that one. http://en.wikipedia.org/wiki/Weierstrass_function – user71352 Jul 03 '14 at 06:00
  • one thing may be I am not getting how to show that it is not monotone? – Topology Jul 03 '14 at 06:02
  • I think this result can be found in Classics on Fractals. – user71352 Jul 03 '14 at 06:12
  • Fits my expectation. Continuity of the function is not strong enough, continuity of the derivative would be good. But we do not have that. – mvw Jul 03 '14 at 06:16
  • @mvw You may find it interesting that even the assumption of differentiability is not quite enough. http://www.ams.org/journals/proc/1976-056-01/S0002-9939-1976-0396870-2/S0002-9939-1976-0396870-2.pdf – user71352 Jul 03 '14 at 06:54
  • Very interesting, but I lack the tools used there. Funny is "by pushing and crushing it is not hard to prove..", which seems to refer to some technique I do not know. – mvw Jul 03 '14 at 07:06
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    Since no one seems to have said the name, Lebesgue's differentiation theorem says that a monotone function is differentiable a.e. (See Royden for example). So any nowhere-differentiable function is non-monotone on any subinterval. – InTransit Jul 03 '14 at 07:37
  • Good point InTransit! – user71352 Jul 03 '14 at 07:42
  • User71352 many thnx for posting the link for the Weil paper. The bare statement of the theorem is mindboggling and audacious! – InTransit Jul 03 '14 at 07:48
  • @InTransit You're welcome. – user71352 Jul 03 '14 at 07:58