We consider the continuous
W
2
,
d
W^{2,d}
-immersions of
d
d
-dimensional hypersurfaces in
R
d
+
1
\mathbb {R}^{d+1}
with second fundamental forms uniformly bounded in
L
d
L^d
. Two results are obtained: first, we construct a family of such immersions whose limit fails to be an immersion of a manifold. This addresses the endpoint cases in J. Langer [Math. Ann. 270 (1985), pp. 223–234], and P. Breuning [J. Geom. Anal. 25 (2015), pp. 1344–1386]. Second, under the additional assumption that the Gauss map is slowly oscillating, we prove that any family of such immersions subsequentially converges to a set locally parametrised by Hölder functions.