  We present three examples to illustrate that in the continuation of a family
of normally hyperbolic $C^1$ manifolds, the normal hyperbolicity may break down
as the continuation parameter approaches a critical value even though the
corresponding generalized Lyapunov-type numbers remain uniformly bounded below
their critical values throughout the process. In the first example, a $C^1$
manifold still exists at the critical parameter value, but it is no longer
normally hyperbolic. In the other two examples, at the critical parameter value
the family of $C^1$ manifolds converges to a nonsmooth invariant set, for which
generalized Lyapunov-type numbers are undefined.
