Through self-study, I found out that there are some pathological functions that do not have subgradients. https://www.stat.cmu.edu/~siva/teaching/725/lec2.pdf
On page 2:
Except for some very pathological functions (and only at the boundary of their domain) subgradients always exist
What is an example of this? And why would it not have a subgradient?