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 ? 



Is a convex function continuous and almost everywhere differentiable?
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".


