  The existence of non-continuous invariant graphs (or strange non-chaotic
attractors) in quasiperiodically forced systems has generated great interest,
but there are still very few rigorous results about the properties of these
objects. In particular, it is not known whether the topological closure of such
graphs is typically a filled-in set, i.e consists of a single interval on every
fibre, or not. We give a positive answer to this question for the class of
so-called pinched skew products, where non-continuous invariant graphs occur
generically, provided the rotation number on the base is diophantine and the
system satisfies some additional conditions. For typical parameter families
these conditions translate to a lower bound on the parameter. On the other
hand, we also construct examples, where the non-continuous invariant graphs
contain isolated points, such that their topological closure cannot be filled
in.
