I'm told that Rockafellar gave an example of a real extended function defined on a locally convex space, whose subdifferential is empty at each point of its domain. The function is proper, lower semicontinuous and convex. Of course this function cannot be defined on a Banach space, becuase we know that in that case the domain of the subdifferential is dense on the domain of the function, and hence nonempty.
Where can I find this example?
A. Br¢ndsted and R. T. Rockafellar,"On the subdifferentiability of convex functions", Bull. Amer. Math. Soc. 16 (1965), 605-611. (see section 5)