Are there any apparently easy problems or conjectures in basic real analysis (that is, calculus) that are still open? By apparently easy, I mean: so much so, that, if it was for the statement alone, they could be part of a calculus book for undergraduates?
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Look for questions marked as open-problem or open-problem-list on mathoverflow. I guess you will find some open problems there.
Here a list of some open questions i have found there:
Stephan Kulla
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http://en.wikipedia.org/wiki/Invariant_subspace_problem
– Christopher A. Wong Jan 07 '15 at 23:12