Here, A and B are respectively vector and scalar Borel functions defined on t2 x R 1 x R ~, where f2 is a bounded, open connected subset of Euclidean space RL The symbol Vu denotes the gradient of u=u(x) , where x = (xl . . . . . x~). Under certain structural assumptions on the coefficients of (1) it has been shown by several authors that a weak solution is H61der continuous in I2; cf. [LU], [S], IT]. The proof in [LU] relies on techniques introduced by DEGIORGI in his investigation of weak solutions of linear elliptic equations in divergence form [DG], whereas the proofs given in [S] and [T] are based on MOSER'S iteration method, [MO]. The main purpose of this paper is to show that if the upper capacitary density of R " f 2 at a point XoeSf2 is positive, then a bounded solution of (1) which takes on continuous boundary values in a weak sense assumes its value at x o continuously. The capacity used here is one whose null sets are the exceptional sets for Sobolev functions. In particular, if the capacity of a set is zero, then its Hausdorff dimension is at most n p , where p is a number determined by the structural assumptions. (In case p--2 , this capacity is equivalent to the classical Newtonian one.) It is also shown that the solution of (1) has an approximate limit at all points of the boundary of f2. Our proof employs the method of DEGIORGI, as presented in [LU].