Differentiability is the existence of a derivative at a point. For maps between finite-dimensional normed vector spaces, the existence of a Frechet derivative requires a single linear map approximating the increment with error . Merely having partial derivatives need not provide this approximation; continuous partial derivatives on a neighbourhood do suffice.
A smooth function is flat at when every derivative, including order zero, vanishes at . The function off zero, extended by zero at zero, is flat there but not identically zero near zero. Thus its Taylor series does not determine it. A real analytic function that is flat at a point vanishes on a neighborhood of that point.
A smooth cutoff function is a smooth function used to localize an argument, typically equal to one on one region and zero outside a slightly larger region. Such functions can be built from bump functions and smooth partitions of unity.
A bump function is a smooth function with compact support. Rescaled and translated bump functions provide local cutoffs.
A function can be differentiable even when its partial derivatives are discontinuous. For example,away from the origin, with value zero at the origin, is differentiable there while its partial derivatives oscillate along the coordinate axes.
