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On the other hand, given a countable union of closed sets $E$, we construct a continuous function $f$ such that $\\o"},"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":"1708.08220","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-08-28T07:33:25Z","cross_cats_sorted":[],"title_canon_sha256":"153a5717054e19914959b7faafb15c57defba0e8889c42267b0636582991e445","abstract_canon_sha256":"c37ba7d3c9992832416a2099ba0889da1676cecaa2ab273a0478c6566f3b3232"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:54.626684Z","signature_b64":"TQFdpKYN093WUxuGda3KMveYBanz0axW4ep8zVdR70ul4AXeT6nqA82UtC/I4CC7r90SwjOvguPAt3QyU6cSBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b1f552acb04ac764bf901615340e2e5d752adf1e00cfd2ae341937996ca4cc83","last_reissued_at":"2026-05-18T00:14:54.625920Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:54.625920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On sets where $\\operatorname{lip} f$ is finite","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Bruce Hanson, Martin Rmoutil, Thomas Z\\\"urcher, Zolt\\'an Buczolich","submitted_at":"2017-08-28T07:33:25Z","abstract_excerpt":"Given a function $f\\colon \\mathbb{R}\\to \\mathbb{R}$, the so-called \"little lip\" function $\\operatorname{lip} f$ is defined as follows: \\begin{equation*} \\operatorname{lip}\n  f(x)=\\liminf_{r{\\scriptscriptstyle \\searrow} 0}\\sup_{|x-y|\\le r} \\frac{|f(y)-f(x)|}{r}. \\end{equation*} We show that if $f$ is continuous on $\\mathbb{R}$, then the set where $\\operatorname{lip} f$ is infinite is a countable union of a countable intersection of closed sets (that is an $F_{\\sigma \\delta}$ set). On the other hand, given a countable union of closed sets $E$, we construct a continuous function $f$ such that $\\o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.08220","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1708.08220","created_at":"2026-05-18T00:14:54.626013+00:00"},{"alias_kind":"arxiv_version","alias_value":"1708.08220v2","created_at":"2026-05-18T00:14:54.626013+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.08220","created_at":"2026-05-18T00:14:54.626013+00:00"},{"alias_kind":"pith_short_12","alias_value":"WH2VFLFQJLDW","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_16","alias_value":"WH2VFLFQJLDWJP4Q","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_8","alias_value":"WH2VFLFQ","created_at":"2026-05-18T12:31:53.515858+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WH2VFLFQJLDWJP4QCYKTIDROLV","json":"https://pith.science/pith/WH2VFLFQJLDWJP4QCYKTIDROLV.json","graph_json":"https://pith.science/api/pith-number/WH2VFLFQJLDWJP4QCYKTIDROLV/graph.json","events_json":"https://pith.science/api/pith-number/WH2VFLFQJLDWJP4QCYKTIDROLV/events.json","paper":"https://pith.science/paper/WH2VFLFQ"},"agent_actions":{"view_html":"https://pith.science/pith/WH2VFLFQJLDWJP4QCYKTIDROLV","download_json":"https://pith.science/pith/WH2VFLFQJLDWJP4QCYKTIDROLV.json","view_paper":"https://pith.science/paper/WH2VFLFQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1708.08220&json=true","fetch_graph":"https://pith.science/api/pith-number/WH2VFLFQJLDWJP4QCYKTIDROLV/graph.json","fetch_events":"https://pith.science/api/pith-number/WH2VFLFQJLDWJP4QCYKTIDROLV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WH2VFLFQJLDWJP4QCYKTIDROLV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WH2VFLFQJLDWJP4QCYKTIDROLV/action/storage_attestation","attest_author":"https://pith.science/pith/WH2VFLFQJLDWJP4QCYKTIDROLV/action/author_attestation","sign_citation":"https://pith.science/pith/WH2VFLFQJLDWJP4QCYKTIDROLV/action/citation_signature","submit_replication":"https://pith.science/pith/WH2VFLFQJLDWJP4QCYKTIDROLV/action/replication_record"}},"created_at":"2026-05-18T00:14:54.626013+00:00","updated_at":"2026-05-18T00:14:54.626013+00:00"}