TITLE: An empty subdifferential
QUESTION [2 upvotes]: Can you give me an example of function $f$ defined on an Hilbert space, real valued (extended with $+ \infty$), lower semi continuous, convex and proper for which $\operatorname{dom}(\partial f)= \emptyset$? 
For $\operatorname{dom}(\partial f)$ I mean the domain of subdifferentiability of $f$.

REPLY [0 votes]: Under your assumptions the subdifferential $\partial f$ is a maximal monotone operator so it cannot be empty.
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