Approximation of arbitrary convex function I am currently studying Jürgen Moser's "A New Proof of de Giorgi's Theorem Concerning the Regularity Problem for Elliptic Differential Equations". During the proof of the first lemma, he claims that an arbitrary convex function $f$ can be approximated by a sequence of twice continously differentiable convex functions $f_m$, such that $f_m''(u)=0$ for large $|u|$, $f_m \rightarrow f$ and $f_m'(u) \rightarrow f'(u)$, where $f'(u)$ exists.
Having looked in several books about convex analysis (Rockafellar), I found out that a convex function should be twice differentiable almost everywhere. I further found in one of Rockafellar's books (convex analysis, thm. 25.7) a Theorem that shows the claim, except for the restriction on $f''$. 
Now the question is how can we come up with that restriction on $f''$ and maybe someone has a reference for a book as well, since I would be interested in seeing the proof.
 A: So I figured it out a few days ago and wanted to share it with you.
Most importantly it is a well known result that an arbitrary convex function can be uniformly approximated by a convex $C^{\infty}$-function on any closed bounded subinterval of the domain. I personally am not interested in the construction, but if you are, you might find some answers in the later mentioned references. Since this Approximation is applied on the Lemma of Fatou and the Lemma of Lebesgue, pointwise convergence of the Approximation is actually enough.
Assume $g$ is the function which approximates our convex function $f$ uniformly on the set $[-n,n]$. Then define
$\tilde{g}_n:= \mathbb{1}_{(-\infty,-n)} (g_n(-n)+g_n'(-n)(x+n)) + \mathbb{1}_{[-n,n]} g_n +\mathbb{1}_{(n,\infty)}(g_n(n)+g_n'(n)(x-n))$.
This function then approximates the convex function $f$ pointwise, and the second derivative vanishes outside the bounded set $[-n,n]$.
Some good references on approximation of convex functions might be:
D. Azagra, “Global and fine approximation of convex functions,” Proc. London Math. Soc., vol. 107, p. 799–824, 2013
A. D. Alexandroff, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State University Annals 
J. J. Koliha, "APPROXIMATION OF CONVEX FUNCTIONS", Real Analysis Exchange, Vol. 29(1), 2003/2004, pp. 465–471
R. Tyrrell Rockafellar, "SECOND-ORDER CONVEX ANALYSIS", Journal of Nonlinear and Convex Analysis 1 (1999), 1-16
