Local Integration of Prox-Regular Functions in Hilbert Spaces

We show that prox-regular functions are locally uniquely determined by their subgradients i.e. if two functions are prox-regular at $\bar{x}$ for $\bar{v}$, then in a neighborhood of $(\bar{x}, \bar{v})$, the functions differ by an additive constant. The class of prox-regular functions includes all convex functions, all qualified convexly composite functions (i.e. with an appropriate constraint qualification) and all pln functions. This result represents an improvement over previous results since the class of prox-regular functions is strictly bigger than the class of pln functions (an example is provided in this paper).