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Denote the image \\begin{equation*} f_{U}(C,C)=\\{f(x,y):(x,y)\\in (C\\times C)\\cap U\\}. \\end{equation*} If $\\partial _{x}f$, $\\partial _{y}f$ are continuous on $U,$ and there is a point $(x_{0},y_{0})\\in (C\\times C)\\cap U$ such that \\begin{equation*} 1<\\left\\vert \\frac{\\partial _{x}f|_{(x_{0},y_{0})}}{\\partial _{y}f|_{(x_{0},y_{0})}}\\right\\vert <3\\text{ or }1<\\left\\vert \\frac{\\partial _{y}f|_{(x_{0},y_{0})}}{\\partial _{x}f|_{(x_{0},y_{0})}}\\right\\vert <3, \\end{equation*} then $f"},"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":"1809.01880","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-09-06T08:42:48Z","cross_cats_sorted":["math.MG","math.NT"],"title_canon_sha256":"8aa393265b181aad176013175f1801a206bf07c48b5b28196d67f4b245aef7ab","abstract_canon_sha256":"530ef37347431ef2a61cb63269d5690ef553adc65ca855c09a34529acf857d88"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:21.742262Z","signature_b64":"obaQqqy//NzOyzyZQsWD2Ps8xjojsOSg772ASuP9lGz3S/er5cXrYTqUPJye92/T0V7d1iSWtS8qN0m5u2OPAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0fb376cd8cbcf67cbc88c5c85a82373c9703e78ff836cb239358f0436506bdb9","last_reissued_at":"2026-05-18T00:06:21.741867Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:21.741867Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Interiors of continuous images of the middle-third Cantor set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.NT"],"primary_cat":"math.DS","authors_text":"Kan Jiang, Lifeng Xi","submitted_at":"2018-09-06T08:42:48Z","abstract_excerpt":"Let $C$ be the middle-third Cantor set, and $f$ a continuous function defined on an open set $U\\subset \\mathbb{R}^{2}$. Denote the image \\begin{equation*} f_{U}(C,C)=\\{f(x,y):(x,y)\\in (C\\times C)\\cap U\\}. \\end{equation*} If $\\partial _{x}f$, $\\partial _{y}f$ are continuous on $U,$ and there is a point $(x_{0},y_{0})\\in (C\\times C)\\cap U$ such that \\begin{equation*} 1<\\left\\vert \\frac{\\partial _{x}f|_{(x_{0},y_{0})}}{\\partial _{y}f|_{(x_{0},y_{0})}}\\right\\vert <3\\text{ or }1<\\left\\vert \\frac{\\partial _{y}f|_{(x_{0},y_{0})}}{\\partial _{x}f|_{(x_{0},y_{0})}}\\right\\vert <3, \\end{equation*} then $f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.01880","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1809.01880","created_at":"2026-05-18T00:06:21.741934+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.01880v1","created_at":"2026-05-18T00:06:21.741934+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.01880","created_at":"2026-05-18T00:06:21.741934+00:00"},{"alias_kind":"pith_short_12","alias_value":"B6ZXNTMMXT3H","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_16","alias_value":"B6ZXNTMMXT3HZPEI","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_8","alias_value":"B6ZXNTMM","created_at":"2026-05-18T12:32:13.499390+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/B6ZXNTMMXT3HZPEIYXEFVARXHS","json":"https://pith.science/pith/B6ZXNTMMXT3HZPEIYXEFVARXHS.json","graph_json":"https://pith.science/api/pith-number/B6ZXNTMMXT3HZPEIYXEFVARXHS/graph.json","events_json":"https://pith.science/api/pith-number/B6ZXNTMMXT3HZPEIYXEFVARXHS/events.json","paper":"https://pith.science/paper/B6ZXNTMM"},"agent_actions":{"view_html":"https://pith.science/pith/B6ZXNTMMXT3HZPEIYXEFVARXHS","download_json":"https://pith.science/pith/B6ZXNTMMXT3HZPEIYXEFVARXHS.json","view_paper":"https://pith.science/paper/B6ZXNTMM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.01880&json=true","fetch_graph":"https://pith.science/api/pith-number/B6ZXNTMMXT3HZPEIYXEFVARXHS/graph.json","fetch_events":"https://pith.science/api/pith-number/B6ZXNTMMXT3HZPEIYXEFVARXHS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/B6ZXNTMMXT3HZPEIYXEFVARXHS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/B6ZXNTMMXT3HZPEIYXEFVARXHS/action/storage_attestation","attest_author":"https://pith.science/pith/B6ZXNTMMXT3HZPEIYXEFVARXHS/action/author_attestation","sign_citation":"https://pith.science/pith/B6ZXNTMMXT3HZPEIYXEFVARXHS/action/citation_signature","submit_replication":"https://pith.science/pith/B6ZXNTMMXT3HZPEIYXEFVARXHS/action/replication_record"}},"created_at":"2026-05-18T00:06:21.741934+00:00","updated_at":"2026-05-18T00:06:21.741934+00:00"}