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We give a partial characterization of these sets in terms of the binary expansion of x. As an application, we consider the differentiability of the composition of Takagi's nowhere differentiable function and the inverse of Lebesgue's singular function."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1012.5535","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-12-26T18:29:18Z","cross_cats_sorted":[],"title_canon_sha256":"414330a63c68d5bdab44eac591acfb52e3c321ff5d0f7dd57a8d5303cc667505","abstract_canon_sha256":"b8ddbe1476c2a9c28896571a21ae5a9f4783ebec998d322dc31963c73b038805"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:32:23.117953Z","signature_b64":"TzHsUXToq2htmB5dBPIjPNcCN9lqY8V79cqG2kTbe2H8vgrP4M08ANQVHMwujW2jumY229IL0M7z5w2qChjDAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bbceb6e78292d19744a70180ba12647ccd78ee6042b299da71f759fd35d547c0","last_reissued_at":"2026-05-18T04:32:23.117061Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:32:23.117061Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"At which points exactly has Lebesgue's singular function the derivative zero ?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Kiko Kawamura","submitted_at":"2010-12-26T18:29:18Z","abstract_excerpt":"Let L_a(x) be Lebesgue's singular function with a real parameter a (0<a<1, a not equal to 1/2). 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