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Corollary 9 in here (page 31) states that a proper convex function $g:Yrightarrow mathbb{R}cup{infty}$ (not necessarily continuous) on a locally convex space $Y$ is subdifferentiable on a point $y$ in the quasi-relative interior of its domain whenever $(y,g(y))$ does not belong to the quasi-relative interior of the epigraph of $g$.

I was wondering when it is possible to omit the last condition about $(y,g(y))$ and still have the subdifferentiablility at $y$.

In particular I may assume one or more of the following:

• 
  $Y$ is a dual space of a Banach space.


• 
  $g$ is lower semicontinuous.


• 
  $g$ is sublinear.


• 
  $g$ is continuous on its domain.

You need additional assumptions: Let $Y$ be an infinite dimensional Banach space and let $g : V to mathbb{R}$ be a linear, unbounded functional. Then it is clear that $g$ has an empty subdifferential everywhere.

If you add continuity (in $y$), then $partial g(y)$ is not empty (see, e.g., the book by Bauschke&Combettes, Prop. 16.14). However, I feel that this already gives you the condition on $(y,g(y))$.