  We show that in a certain subfamily of the Kuranishi family of any half Inoue
surface the algebraic dimensions of the fibers jump downwards at special points
of the parameter space showing that the upper semi-continuity of algebraic
dimensions in any sense does not hold in general for families of compact
non-Kaehler manifolds. In the Kaehler case, the upper semi-continuity always
holds true in a certain weak sense.
