On a Non-Prox-Regularity Example by Rockafellar and Wets

We analyze a counterexample concerning non-prox-regular functions provided by Rockafellar and Wets in their 1998 foundational monograph, $\mathit{\text{Variational}}\mathit{\text{Analysis}}$. By establishing a general theorem on the continuity of set-valued mappings, we show that the counterexample does not have its intended properties. We propose a slight adjustment to the construction and prove that the corrected version serves as the desired counterexample.