The broad class of extended real-valued lower semicontinuous lsc functions on Rn captures nearly all functions of practical importance in equation solving, variational problems, fitting, and estimation. The paper develops piecewise polynomial functions, called epi-splines, that approximateany lsc function to an arbitrary level of accuracy. Epi-splines provide the foundation for the solution of a rich class of function identification problems that incorporate general constraints on the function to be identified including those derived from information about smoothness, shape, proximity to other functions, and so on. As such extrinsic information as well as observed function and subgradient values often evolve in applications, we establish conditions under which the computed epi-splines converge to the function we seek to identify. Numerical examples in response surface building and probability density estimation illustrate the framework.