The quality of a finite element solution hinges in particular on the approximation properties of the finite element space.  In the first part of this talk we will consider the approximation of the gradient of a target function by continuous piecewise polynomials over a simplicial, 'shape-regular' mesh and prove the following result: the global best approximation error is equivalent to an appropriate sum in terms of the local best approximation errors on the elements, which do not overlap.  This means in particular that, for gradient norms, the continuity requirement does not downgrade the local approximation potential on elements and that discontinuous piecewise polynomials do not offer additional approximation power.  In the second part of the talk we will discuss the usefulness of this result in the context of adaptive methods for partial differential equations. Joint work with Francesco Mora (Milan).