Can a smooth convex functions be non-differentiable? Consider the definition of the $\beta$-smoothness (for some constant $\beta$): 
$$
\|\left. \nabla f \right|_{ y } - \left. \nabla f \right|_{ x }  \| \leq \beta \| x - y \| 
$$
And convexity: 
$$
f(x) \geq f(y)+  \left. \nabla f \right|_{ y } . (x - y), \forall x, y
$$
Can a smooth convex function be non-differentiable at some points on its domain? (and why?)
In the definitions, $\left. \nabla f \right|_{y}$ is subgradient of the function $f$ at point $y$, if it is not differentiable at this point; so the definitions of convexity and smoothness hold even for non-differentiable functions. 
 A: A convex function $f$ is non-differentiable at a point $x$ iff the subgradient $\nabla f|_x$ has more than one vector.  I presume the inequality
$\|\nabla f|_x - \nabla f|_y\| \le \beta \|x - y \|$  means 
$\|v - w\| \le \beta \|x - y\|$ for all $v \in \nabla f|_x$ and $w \in \nabla f|_y$.  In particular, for $y = x$ you see that $\nabla f|_x$ must contain only one vector for this to be true.
A: I presume that the definition of smooth-ness you are using is something specific to convex functions. Its not the standard definition of a function that a function is continuously differentiable and has derivatives of all orders C∞; 
https://en.wikipedia.org/wiki/Smoothness although I note that you are using a particular definition here. 
I note that discrete analogues involving subgradients and the like are also used in characterizing (not necessarily diff-erentiable, in the traditional sense) 'pseudo convex functions' using more discrete notion such as the "dini 'derivative" see https://en.wikipedia.org/wiki/Dini_derivative
