Cute undergraduate real analysis problem

Asked Dec 20 '12 at 02:10

Active Dec 20 '12 at 02:19

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Possible Duplicate:
Showing range is countable

I was going through my old notebooks and found this. I thought it was good enough to share.

Problem: Let $f(x)$ be a function $\mathbb R \rightarrow \mathbb R$, with the only restriction on it being that it has a local extremum at each point. Prove it has countable range.

edited Apr 13 '17 at 12:20

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asked Dec 20 '12 at 02:10

Potato

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Bravo! Great poser! – ncmathsadist Dec 20 '12 at 02:19
7

So, was this problem brought to your attention by a cute undergraduate? – Will Jagy Dec 20 '12 at 02:20
@WillJagy Unfortunately not :(. – Potato Dec 20 '12 at 02:21
This is similar to the question Showing range is countable. I actually voted to close as duplicate, but perhaps hastily as here it is a little more general, assuming extremum rather than local minimum. Shai Covo's answer there is on this generalization. – Jonas Meyer Dec 20 '12 at 02:38
@Jonas: one of the answers in the question you linked answers the case of arbitrary extrema (and gives references, too). – Martin Argerami Dec 20 '12 at 04:04
It is literally the same proof. You can do the proof for local minimums from the link, then do the proof for maximums (which is the same). Then notice the range is a subset of the union of two countable sets. Further, if this function is continuous, it is constant. – toypajme Dec 20 '12 at 04:13

Cute undergraduate real analysis problem

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