{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T05:41:46Z","timestamp":1757310106008,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2020,2,14]],"date-time":"2020-02-14T00:00:00Z","timestamp":1581638400000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,2,14]],"date-time":"2020-02-14T00:00:00Z","timestamp":1581638400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,2,14]],"date-time":"2020-02-14T00:00:00Z","timestamp":1581638400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,4,1]]},"abstract":"<jats:p>Following Chaudhuri, Sankaranarayanan, and Vardi, we say that a function $f:[0,1] \\to [0,1]$ is $r$-regular if there is a B\\&amp;quot;{u}chi automaton that accepts precisely the set of base $r \\in \\mathbb{N}$ representations of elements of the graph of $f$. We show that a continuous $r$-regular function $f$ is locally affine away from a nowhere dense, Lebesgue null, subset of $[0,1]$. As a corollary we establish that every differentiable $r$-regular function is affine. It follows that checking whether an $r$-regular function is differentiable is in $\\operatorname{PSPACE}$. Our proofs rely crucially on connections between automata theory and metric geometry developed by Charlier, Leroy, and Rigo.<\/jats:p>","DOI":"10.23638\/lmcs-16(1:17)2020","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:45:11Z","timestamp":1743702311000},"source":"Crossref","is-referenced-by-count":1,"title":["Continuous Regular Functions"],"prefix":"10.23638","volume":"Volume 16, Issue 1","author":[{"given":"Alexi Block","family":"Gorman","sequence":"first","affiliation":[]},{"given":"Philipp","family":"Hieronymi","sequence":"additional","affiliation":[]},{"given":"Elliot","family":"Kaplan","sequence":"additional","affiliation":[]},{"given":"Ruoyu","family":"Meng","sequence":"additional","affiliation":[]},{"given":"Erik","family":"Walsberg","sequence":"additional","affiliation":[]},{"given":"Zihe","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Ziqin","family":"Xiong","sequence":"additional","affiliation":[]},{"given":"Hongru","family":"Yang","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2020,2,14]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1901.03366v3","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1901.03366v3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:45:11Z","timestamp":1743702311000},"score":1,"resource":{"primary":{"URL":"http:\/\/lmcs.episciences.org\/5301"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,14]]},"references-count":0,"URL":"https:\/\/doi.org\/10.23638\/lmcs-16(1:17)2020","relation":{"has-preprint":[{"id-type":"arxiv","id":"1901.03366v2","asserted-by":"subject"},{"id-type":"arxiv","id":"1901.03366v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1901.03366","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1901.03366","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2020,2,14]]},"article-number":"5301"}}