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I am hoping that this question (if in acceptable form) be community wiki.

Are there any open conjectures in real analysis primarily targeting real valued functions of a single real variable ? (it may involve other concepts but primarily targeting these type of functions and no involvement of number theory). Its not that i am going to attack them right away ( not due to lack of interest but time) but am just curious to know them. Also how difficult it is get a new result in this area ?

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    $\begingroup$ I hope this gets some interesting answers! $\endgroup$ – Olivier Bégassat Aug 20 '11 at 13:07
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    $\begingroup$ Why did you accept an answer already? There probably are many more open problems. $\endgroup$ – Jonas Teuwen Aug 21 '11 at 16:25
  • $\begingroup$ @Jonas Teuwen : The current answer by Gerry Myerson gives some pertinent examples. I have just accepted, but if by doing so it prevents others from giving more answers ( may be by not appearing on the main page or some such thing), I'd rather wait. But i really do not understand how to select answer in case of mutiple answers giving suitable examples as this question itself is not very specific. More over if you say that it should be done based on the importance of the examples then probably i am not qualified to do such a thing. I hope there would be a discussion which takes care of it. $\endgroup$ – Rajesh Dachiraju Aug 21 '11 at 16:42
  • $\begingroup$ I agree with Jonas that you shouldn't be so quick to accept an answer. Wait until things settle down. As to which answer to accept if/when there are many answers given, just accept the one that you personally find most useful. Your vote will be read that way, and not as a judgement from on high as to which answer presents the most important examples. $\endgroup$ – Gerry Myerson Aug 24 '11 at 1:13
  • $\begingroup$ It would appear that things have settled down. Maybe it's time to accept an answer. $\endgroup$ – Gerry Myerson Oct 25 '13 at 10:08
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Probably the quickest way to find a large number of problems of current interest in real analysis is to look through the journal "Real Analysis Exchange", especially the Queries section and the open problems listed in (most of) the survey articles.

(added October 28) Brian S. Thomson's book Symmetric Properties of Real Functions (Marcel Dekker, 1994) has several pages of problems at the end. Some of these problems have been open for a long time. Some of the other problems were simply questions that occurred to Thomson when he wrote the book and wasn't able to answer right off. Many of these problems have been answered by now, but I'm sure there remain many that are still unanswered. Since the topic of this book is rather specialized, the number and variety of problems serves well to indicate there is no lack of open problems targeting real valued functions of a real variable.

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Khabibullin’s conjecture on integral inequalities. Also, according to this entry at the Open Problem Garden, the following question is open: Give a necessary and sufficient criterion for the sequence $a_n$ so that the power series $\sum a_nx^n$ is bounded for all $x$ in $\bf R$.

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