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We consider two natural versions of the continuity, which we call strong and weak, of $f_A^{-1}$ (which is of course multi-valued) on the field of values $F(A)$. The strong continuity holds, in particular, on the interior of $F(A)$, and at such points $z \\in \\partial F(A)$ which are either corner points, belong to the relative interior of flat portions of $\\partial F(A)$, or whose preimage under $f_A$ is contained in a one-dimensional set. Consequently, $f_A^{-1}$ is cont"},"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":"1307.5033","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-07-18T18:27:10Z","cross_cats_sorted":[],"title_canon_sha256":"e7c868ec615669be593aaaa824de7c4dcd1d5d88ad303503a41e1e2d7e038121","abstract_canon_sha256":"966835c8e3f58ac351b41dd6d2d95e05ba3926c99d55e7165eaf5d712f9cbd1b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:06.875344Z","signature_b64":"RiqGy64hBsl/pU3ucJrEj7EnRZdpBGa9QOWOUsYkA0kKxk3hnUyM+TGEBsSQLVRMxtLvKyKN5QI/c6PtxuTVAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b6463c508e2201f4c046f92f1e4391b1ba7fc36c66c581ffdb9205382549a517","last_reissued_at":"2026-05-18T03:18:06.874515Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:06.874515Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Continuity properties of vectors realizing points in the classical field of values","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Brian Lins, Charles R. Johnson, Dan Corey, Ilya Spitkovsky, Ryan Kirk","submitted_at":"2013-07-18T18:27:10Z","abstract_excerpt":"For an $n$-by-$n$ matrix $A$, let $f_A$ be its \"field of values generating function\" defined as $f_A\\colon x\\mapsto x^*Ax$. We consider two natural versions of the continuity, which we call strong and weak, of $f_A^{-1}$ (which is of course multi-valued) on the field of values $F(A)$. The strong continuity holds, in particular, on the interior of $F(A)$, and at such points $z \\in \\partial F(A)$ which are either corner points, belong to the relative interior of flat portions of $\\partial F(A)$, or whose preimage under $f_A$ is contained in a one-dimensional set. 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