  In the study of discontinuous groups for non-Riemannian homogeneous spaces,
the idea of "continuous analogue" gives a powerful method (T. Kobayashi [Math.
Ann. 1989]). For example, a semisimple symmetric space G/H admits a
discontinuous group which is not virtually abelian if and only if G/H admits a
proper SL(2,R)-action (T. Okuda [J. Differential Geom. 2013]). However, the
action of discrete subgroups is not always approximated by that of connected
groups. In this paper, we show that the theorem cannot be extended to general
homogeneous spaces G/H of reductive type. We give a counterexample in the case
G = SL(5,R).
