  This is the first of a series of papers that develop a systematic bridge
between constructions in discrete mathematics and the corresponding continuous
analogs. In this paper, we establish an equivalence between Forman's discrete
Morse theory on a simplicial complex and the continuous Morse theory (in the
sense of any known non-smooth Morse theory) on the associated order complex via
the Lov\'asz extension. Furthermore, we propose a new version of the
Lusternik-Schnirelman category on abstract simplicial complexes to bridge the
classical Lusternik-Schnirelman theorem and its discrete analog on finite
complexes. More generally, we can suggest a discrete Morse theory on
hypergraphs by employing piecewise-linear (PL) Morse theory and Lov\'asz
extension, hoping to provide new tools for exploring the structure of
hypergraphs.
