The focus of this paper is on ``flexible local approximation'' and on methods capable of proving it, employing a variety of approximating functions not at all limited to polynomials. New ideas are considered. A new class of flexible local approximation methods (FLAME) is introduced, where the difference scheme is defined by the chosen set of local basis functions and the grid stencil. The motivation of developing this class of methods is to minimize the notorious staircase effect at curved and slanted interface boundaries on regular Cartesian grids. As illustrative examples, the paper presents arbitrarily high order 3-point schemes for the 1D Schrödinger equation and a 1D singular equation, schemes for electrostatic interactions of colloidal particles, electromagnetic wave propagation and scattering, and plasmon resonances.