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is this statement true ? assume $f:D\rightarrow \mathbf{R} $ is a convex function where $D\subset \mathbf{R}^n$ is a convex set. $f$ is continuous and almost everywhere differentiable and in class $C^n$.
$C^n$: calss of $n$ time differentiable functions.
if the above statement is true prove it please and if not true, please bring up a counter example.

EDIT: Enrquie has addressed what I was looking for second order differentiability of convex function by bringing up this paper "SECOND ORDER DIFFERENTIABILITY OF CONVEX FUNCTIONS IN BANACH SPACES" . However can any one address higher order (more than second order) differentiability in a.e. sense for convex function too ?

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  • $\begingroup$ What's your definition of a vector valued convex function? $\endgroup$ Oct 5, 2013 at 8:59
  • $\begingroup$ thank you very much for your remark. I have limited the function to be only real valued. $\endgroup$ Oct 5, 2013 at 10:14
  • $\begingroup$ You want this property inside $D$, at the boundary continuity may fail even in one dimension. $\endgroup$ Oct 5, 2013 at 10:46
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    $\begingroup$ Dear berhard mahboobi, what you say is almost true, but: continuous, no (not up to the boundary); a.e. differentiable: yes in R^n, but "a.e." has no or little meaning in infinite dimensional TVS; C^n: no, think to |x|. Please check any textbook on the subject. Note that this site is devoted to research matter. $\endgroup$ Oct 5, 2013 at 10:48

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In an infinite dimensional Banach space, not every norm (or every linear mapping), which are convex functions, are continuous. The Banach spaces in which every continuous convex function defined in a convex open set is differentiable in a dense $G_\delta$ set are called Asplund spaces. A lot of recent research about these spaces is available. Even wikipedia can be useful for this

http://en.wikipedia.org/wiki/Asplund_space

Sorry, I read again and understand that the question means if the function is conntinuous in a, let say, "big subset".

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  • $\begingroup$ @ Enrique, thank you, based on your reference and conclusion, can you please restate my question in a single theorem for more accuracy ? please note that $C^n$ is the class of functions which are n-times differentiable. $\endgroup$ Oct 5, 2013 at 11:47
  • $\begingroup$ Dear behrad, the notion of secon differentiability in a function that only is in principle differentiable in a dense but not open set is not so trivial. I advise you to read this Borwein and Noll paper, and the references therein. And, as other people is saying, your questions are most appropiate in other forums, perhaps mathhelpforum or Mathexchange. ams.org/journals/tran/1994-342-01/S0002-9947-1994-1145959-4/…. $\endgroup$
    – Enrique
    Oct 5, 2013 at 15:51
  • $\begingroup$ The other message perhaps is too rude. I am not user here but I am observing that in this forum even continue this discussion can be inappropiate. If you want to write an email here I can contact you and try to answer your questions. Best wishes $\endgroup$
    – Enrique
    Oct 5, 2013 at 16:29
  • $\begingroup$ Dear Enrique, I wonder what have disturbed you ? non of the comment i see here is not rude and your answer is also nice. I will contact you, however answering this question here might encourage other people to contribute in our correspondence. $\endgroup$ Oct 7, 2013 at 11:31
  • $\begingroup$ Dear Behrad, you didn't disturb me. But I am here accidentally, looking for other information, and I have observed that this is a professional forum and the users are asking to close this question here. Hence, I suppose that continuing the correspondence here and keeping the question active can be not appropiate, in this forum. I seems that there are other forums where the question can be naturally discussed. The topic of differentiability in Banach spaces is really nice and rich, I am sure that a lot of people would like to help you, but I believe that in the other, the Math Exchange.. $\endgroup$
    – Enrique
    Oct 8, 2013 at 11:28

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