  Given two discrete Morse functions on a simplicial complex, we introduce the
{\em connectedness homomorphism} between the corresponding discrete Morse
complexes. This concept leads to a novel framework for studying the
connectedness in discrete Morse theory at the chain complex level. In
particular, we apply it to describe a discrete analogy to `cusp-degeneration'
of Morse complexes. A precise comparison between smooth case and our discrete
cases is also given.
