{"paper":{"title":"Differentiability of a two-parameter family of self-affine functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Pieter C. Allaart","submitted_at":"2016-06-24T21:34:52Z","abstract_excerpt":"This paper highlights an unexpected connection between expansions of real numbers to noninteger bases (so-called {\\em $\\beta$-expansions}) and the infinite derivatives of a class of self-affine functions. Precisely, we extend Okamoto's function (itself a generalization of the well-known functions of Perkins and Katsuura) to a two-parameter family $\\{F_{N,a}: N\\in\\mathbb{N}, a\\in(0,1)\\}$. We first show that for each $x$, $F_{N,a}'(x)$ is either $0$, $\\pm\\infty$, or undefined. We then extend Okamoto's theorem by proving that for each $N$, depending on the value of $a$ relative to a pair of thres"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.07838","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}