  The aim of this paper is to study the convergence and divergence of the
Rogers-Ramanujan and the generalized Rogers-Ramanujan continued fractions on
the unit circle. We provide an example of an uncountable set of measure zero on
which the Rogers-Ramanujan continued fraction $R(x)$ diverges and which
enlarges a set previously found by Bowman and Mc Laughlin. We further study the
generalized Rogers-Ramanujan continued fractions $R_a(x)$ for roots of unity
$a$ and give explicit convergence and divergence conditions. As such, we extend
some work of Huang towards a question originally investigated by Ramanujan and
some work of Schur on the convergence of $R(x)$ at roots of unity. In the end,
we state several conjectures and possible directions for generalizing Schur's
result to all Rogers-Ramanujan continued fractions $R_a(x)$.
