We investigate the problem of incremental learning in artificial neural networks by viewing it as a sequential function approximation problem. A framework for discussing the generalization ability of a trained network in the original function space using tools of functional analysis based on reproducing kernel Hilbert spaces (RKHS) is introduced. Using this framework, we devise a method of carrying out optimal incremental learning with respect to the entire set of training data by employing the results derived at the previous stage of learning and incorporating the newly available training data effectively. Most importantly, the incrementally learned function has the same (optimal) generalization ability as would have been achieved by using batch learning on the entire set of training data, hence, referred to as exact learning. This ensures that both the learning operator and the learned function can be computed using an online incremental scheme. Finally, we also provide a simplified closed-form relationship between the learned functions before and after the incorporation of new data for various optimization criteria, opening avenues for work into selection of optimal training set. We also show that learning under this kind of framework is inherently well suited for applying novel model selection strategies and introducing bias and a priori knowledge in a more systematic way. Moreover, it provides a useful hint in performing kernel-based approximations, of which the regularization and SVM networks are special cases, in an online setting.