While there are many open problems in Real Analysis like Khabibullin's conjecture or Lehmer's conjecture, those are big enough to take an expert's life for several years, let alone some graduate student that may haven't had any research done yet.

Are there any problems in Real Analysis which has not been solved yet or otherwise have still some rooms to generalize, on the other hand easy enough to consider it as a project for a research-degree student to finish it in a few years?

Suggestions on this would be very much appreciated!

PS - Migrated from math.SE

Yes.Many people get PhDs in analysis, so looking at their theses will give you examples. $\endgroup$ – Kimball Jun 14 '15 at 12:45