We study the class of functions on Lipschitz-graph domains satisfying a differential-oscillation condition and show that such functions are -approximable. As a consequence, we obtain the quantitative Fatou theorem in the spirit of works, for example, by Garnett 6 and Bortz–Hofmann 1. Such a class contains harmonic functions, as well as non-harmonic ones, for example, nonnegative subharmonic functions whose gradient norm is quasi-nearly subharmonic, as illustrated by our discussion.
Adamowicz et al. (Tue,) studied this question.