  We investigate the continuity properties of the homogenized boundary data
$\overline{g}$ for oscillating Dirichlet boundary data problems. We show that,
for a generic non-rotation-invariant operator and boundary data, $\overline{g}$
is discontinuous at every rational direction. In particular this implies that
the continuity condition of Choi and Kim is essentially sharp. On the other
hand, when this condition holds, we show a H\"{o}lder modulus of continuity for
$\overline{g}$. When the operator is linear we show that $\overline{g}$ is
H\"{o}lder-$\frac{1}{d}$ up to a logarithmic factor. The proofs are based on a
new geometric observation on the limiting behavior of $\overline{g}$ at
rational directions, reducing to a class of two dimensional problems for
projections of the homogenized operator.
