(untitled) (Analysis in Theory and Applications, 2017)
In this paper, we introduce a condition weaker than the L^p differentiability, which we call C^p condition. We prove that if a function satisfies this condition at a point, then there exists the best local approximation at that point. We also give a necessary and sufficient condition for that a function be L^p differentiable. In addition, we study the convexity of the set of cluster points of the net of best approximations of f, {P_ε(f)} as ε → 0.