  In this paper, an upper semismooth function is defined to be a lower
semicontinuous function whose radial subderivative satisfies a mild directional
upper semicontinuity property. Examples of upper semismooth functions are the
proper lower semicontinuous convex functions, the lower-C$^1$ functions, the
regular directionally Lipschitz functions, the Mifflin semismooth functions,
the Thibault-Zagrodny directionally stable functions. It is shown that the
radial subderivative of such functions can be recovered from any
subdifferential of the function. It is also shown that these functions are
subdifferentially determined, in the sense that if two functions have the same
subdifferential and one of the functions is upper semismooth, then the two
functions are equal up to an additive constant.
