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In this paper, we establish an equivalence between Forman\u2019s 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\u00e1sz extension. Furthermore, we propose a new version of the Lusternik\u2013Schnirelman category on abstract simplicial complexes to bridge the classical Lusternik\u2013Schnirelman 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\u00e1sz extension, hoping to provide new tools for exploring the structure of hypergraphs.<\/jats:p>","DOI":"10.1007\/s00454-022-00461-1","type":"journal-article","created":{"date-parts":[[2023,7,21]],"date-time":"2023-07-21T19:01:23Z","timestamp":1689966083000},"page":"49-72","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Discrete-to-Continuous Extensions: Lov\u00e1sz Extension and Morse Theory"],"prefix":"10.1007","volume":"72","author":[{"given":"J\u00fcrgen","family":"Jost","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8439-9156","authenticated-orcid":false,"given":"Dong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,7,21]]},"reference":[{"issue":"3","key":"461_CR1","first-page":"501","volume":"2","author":"P Alexandroff","year":"1937","unstructured":"Alexandroff, P.: Diskrete R\u00e4ume. 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