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A real valued function $f$ on a subset $E$ of $\\R$, the set of real numbers is down continuous if it preserves downward half Cauchy sequences, i.e. the sequence $(f(\\alpha_{n}))$ is downward half Cauchy whenever $(\\alpha_{n})$ is a downward half Cauchy sequence of points in $E$, where a sequence $(\\alpha_{ k})$ of points in $\\R$ is called downward half Cauchy if for every $\\varepsilon>0$ there exists an $n_{0}\\in{\\N}$ such that $\\alpha_{m}-\\alpha_{n} <\\varepsilon$ for $m \\geq n \\geq n_0$. It turns"},"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":"1802.01324","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-02-05T10:10:53Z","cross_cats_sorted":[],"title_canon_sha256":"66ac808225c2d638cd610683ae3bc5a32f8fcaee4ed2573bd95881e09dbae4ea","abstract_canon_sha256":"6773673f27b8113ba9e9baca6fc49701d3223bdb58f48b87caed5bf5e7456138"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:26.721575Z","signature_b64":"wfJ5TflcZOpG5Rx6ZNuWk5u3/0c7qBf8YJ6FmJLwC1vYo+sKcyDoFDqMN1Ju8xG/4uj4o8wQ+4J5nYiNyTRmBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"427e75d6c525e40a104bb5ebc010b260b68d0f48d67dadc092cac04043a74659","last_reissued_at":"2026-05-18T00:24:26.721031Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:26.721031Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A study on downward half Cauchy sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Huseyin Cakalli","submitted_at":"2018-02-05T10:10:53Z","abstract_excerpt":"In this paper, we introduce and investigate the concepts of down continuity and down compactness. A real valued function $f$ on a subset $E$ of $\\R$, the set of real numbers is down continuous if it preserves downward half Cauchy sequences, i.e. the sequence $(f(\\alpha_{n}))$ is downward half Cauchy whenever $(\\alpha_{n})$ is a downward half Cauchy sequence of points in $E$, where a sequence $(\\alpha_{ k})$ of points in $\\R$ is called downward half Cauchy if for every $\\varepsilon>0$ there exists an $n_{0}\\in{\\N}$ such that $\\alpha_{m}-\\alpha_{n} <\\varepsilon$ for $m \\geq n \\geq n_0$. 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