Interpolating Between Choices for the Approximate Intermediate Value Theorem
This paper proves the approximate intermediate value theorem, constructively and from notably weak hypotheses: from pointwise rather than uniform continuity, without assuming that reals are presented with rational approximants, and without using countable choice. The theorem is that if a pointwise c...
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| Format: | Article |
| Language: | English |
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Logical Methods in Computer Science e.V.
2020-07-01
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| Series: | Logical Methods in Computer Science |
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| Online Access: | https://lmcs.episciences.org/2638/pdf |
| _version_ | 1827322733345636352 |
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| author | Matthew Frank |
| author_facet | Matthew Frank |
| author_sort | Matthew Frank |
| collection | DOAJ |
| description | This paper proves the approximate intermediate value theorem, constructively
and from notably weak hypotheses: from pointwise rather than uniform
continuity, without assuming that reals are presented with rational
approximants, and without using countable choice. The theorem is that if a
pointwise continuous function has both a negative and a positive value, then it
has values arbitrarily close to 0. The proof builds on the usual classical
proof by bisection, which repeatedly selects the left or right half of an
interval; the algorithm here selects an interval of half the size in a
continuous way, interpolating between those two possibilities. |
| first_indexed | 2024-04-25T01:34:35Z |
| format | Article |
| id | doaj.art-853992862a6548088c0799f4eff4cc2c |
| institution | Directory Open Access Journal |
| issn | 1860-5974 |
| language | English |
| last_indexed | 2024-04-25T01:34:35Z |
| publishDate | 2020-07-01 |
| publisher | Logical Methods in Computer Science e.V. |
| record_format | Article |
| series | Logical Methods in Computer Science |
| spelling | doaj.art-853992862a6548088c0799f4eff4cc2c2024-03-08T10:31:23ZengLogical Methods in Computer Science e.V.Logical Methods in Computer Science1860-59742020-07-01Volume 16, Issue 310.23638/LMCS-16(3:5)20202638Interpolating Between Choices for the Approximate Intermediate Value TheoremMatthew FrankThis paper proves the approximate intermediate value theorem, constructively and from notably weak hypotheses: from pointwise rather than uniform continuity, without assuming that reals are presented with rational approximants, and without using countable choice. The theorem is that if a pointwise continuous function has both a negative and a positive value, then it has values arbitrarily close to 0. The proof builds on the usual classical proof by bisection, which repeatedly selects the left or right half of an interval; the algorithm here selects an interval of half the size in a continuous way, interpolating between those two possibilities.https://lmcs.episciences.org/2638/pdfmathematics - logiccomputer science - logic in computer science03f60, 03d78, 03e25, 26a15, 26e40g.1.5 |
| spellingShingle | Matthew Frank Interpolating Between Choices for the Approximate Intermediate Value Theorem Logical Methods in Computer Science mathematics - logic computer science - logic in computer science 03f60, 03d78, 03e25, 26a15, 26e40 g.1.5 |
| title | Interpolating Between Choices for the Approximate Intermediate Value Theorem |
| title_full | Interpolating Between Choices for the Approximate Intermediate Value Theorem |
| title_fullStr | Interpolating Between Choices for the Approximate Intermediate Value Theorem |
| title_full_unstemmed | Interpolating Between Choices for the Approximate Intermediate Value Theorem |
| title_short | Interpolating Between Choices for the Approximate Intermediate Value Theorem |
| title_sort | interpolating between choices for the approximate intermediate value theorem |
| topic | mathematics - logic computer science - logic in computer science 03f60, 03d78, 03e25, 26a15, 26e40 g.1.5 |
| url | https://lmcs.episciences.org/2638/pdf |
| work_keys_str_mv | AT matthewfrank interpolatingbetweenchoicesfortheapproximateintermediatevaluetheorem |