  A celebrated result in bifurcation theory is that global connected sets of
non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues
of odd algebraic multiplicity of the linearized problem when the operators
involved are compact. In this paper a simple example is constructed which
satisfies the regularity hypotheses of the global bifurcation theorem, and the
eigenvalue has algebraic multiplicity one, yet all the path-connected
components of the connected sets that bifurcate are singletons. Another example
shows that even when the operators are everywhere infinitely differentiable and
classical bifurcation occurs locally at a simple eigenvalue, the global
continuum may not be path-connected away from the bifurcation point. A third
example shows that the non-trivial solutions which, by variational theory,
bifurcate from eigenvalues of any multiplicity when the problem has gradient
structure, may not be connected and may contain no paths except singletons.
