Abstract: This present paper show that, for general lower semicontinuous functions defined on Hilbert space E,the approximate mean value theorem such as that in[9] is also valid for Proximal-subdiferential. If E=[a,b]⊂R, the mean value theorem in inequality form is obtained. By use of the approximate mean value theorem, the Proximal-subdifferential descriptions of convexity, Lipschitz and constant properties of functions is given.

Key words: Proximal-subdifferential, Generalized mean value theorems, Nonsmooth analysis