A differentiability criterion for continuous functions
We show that, with the exception of the symmetric derivative, each limit of the form lim h → 0 A f ( x + a h ) + B f ( x + b h ) h , ( A + B = 0 , A a + B b = 1 ) , is equivalent to the ordinary derivative, for all continuous functions at x . And, up to a non-zero scalar multiple, these are the only...
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| Veröffentlicht in: | Monatshefte für Mathematik 2022-02, Vol.197 (2), p.285-291 |
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| container_title | Monatshefte für Mathematik |
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| creator | Catoiu, Stefan |
| description | We show that, with the exception of the symmetric derivative, each limit of the form
lim
h
→
0
A
f
(
x
+
a
h
)
+
B
f
(
x
+
b
h
)
h
,
(
A
+
B
=
0
,
A
a
+
B
b
=
1
)
,
is equivalent to the ordinary derivative, for all continuous functions at
x
. And, up to a non-zero scalar multiple, these are the only criteria for differentiating all continuous functions at
x
, by taking limits of first order difference quotients with two function evaluations. |
| doi_str_mv | 10.1007/s00605-021-01574-0 |
| format | Article |
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lim
h
→
0
A
f
(
x
+
a
h
)
+
B
f
(
x
+
b
h
)
h
,
(
A
+
B
=
0
,
A
a
+
B
b
=
1
)
,
is equivalent to the ordinary derivative, for all continuous functions at
x
. And, up to a non-zero scalar multiple, these are the only criteria for differentiating all continuous functions at
x
, by taking limits of first order difference quotients with two function evaluations.</description><identifier>ISSN: 0026-9255</identifier><identifier>EISSN: 1436-5081</identifier><identifier>DOI: 10.1007/s00605-021-01574-0</identifier><language>eng</language><publisher>Vienna: Springer Vienna</publisher><subject>Continuity (mathematics) ; Derivatives ; Mathematics ; Mathematics and Statistics ; Quotients ; Tornadoes</subject><ispartof>Monatshefte für Mathematik, 2022-02, Vol.197 (2), p.285-291</ispartof><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2021</rights><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-24fc8f4627dfd2223eae82ddebe50158b3a8f03040b219f0c2b155117e2f55c23</citedby><cites>FETCH-LOGICAL-c319t-24fc8f4627dfd2223eae82ddebe50158b3a8f03040b219f0c2b155117e2f55c23</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00605-021-01574-0$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00605-021-01574-0$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Catoiu, Stefan</creatorcontrib><title>A differentiability criterion for continuous functions</title><title>Monatshefte für Mathematik</title><addtitle>Monatsh Math</addtitle><description>We show that, with the exception of the symmetric derivative, each limit of the form
lim
h
→
0
A
f
(
x
+
a
h
)
+
B
f
(
x
+
b
h
)
h
,
(
A
+
B
=
0
,
A
a
+
B
b
=
1
)
,
is equivalent to the ordinary derivative, for all continuous functions at
x
. And, up to a non-zero scalar multiple, these are the only criteria for differentiating all continuous functions at
x
, by taking limits of first order difference quotients with two function evaluations.</description><subject>Continuity (mathematics)</subject><subject>Derivatives</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Quotients</subject><subject>Tornadoes</subject><issn>0026-9255</issn><issn>1436-5081</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kE9LxDAQxYMoWFe_gKeC5-jMpEnb47L4Dxa86Dm0aSJZ1mZN2sN-e6MVvHkamHnvzePH2DXCLQLUdwlAgeRAyAFlXXE4YQVWQnEJDZ6yAoAUb0nKc3aR0g4AUKi2YGpdDt45G-04-a73ez8dSxP9ZKMPY-lCLE3Ip3EOcyrdPJop79MlO3PdPtmr37libw_3r5snvn15fN6st9wIbCdOlTONqxTVgxuISNjONjQMtrcy92x60TUOBFTQE7YODPUoJWJtyUlpSKzYzZJ7iOFztmnSuzDHMb_UpAQ2LdQgs4oWlYkhpWidPkT_0cWjRtDffPTCR2c--oePhmwSiyll8fhu41_0P64vgmVnvQ</recordid><startdate>20220201</startdate><enddate>20220201</enddate><creator>Catoiu, Stefan</creator><general>Springer Vienna</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20220201</creationdate><title>A differentiability criterion for continuous functions</title><author>Catoiu, Stefan</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-24fc8f4627dfd2223eae82ddebe50158b3a8f03040b219f0c2b155117e2f55c23</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Continuity (mathematics)</topic><topic>Derivatives</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Quotients</topic><topic>Tornadoes</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Catoiu, Stefan</creatorcontrib><collection>CrossRef</collection><jtitle>Monatshefte für Mathematik</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Catoiu, Stefan</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A differentiability criterion for continuous functions</atitle><jtitle>Monatshefte für Mathematik</jtitle><stitle>Monatsh Math</stitle><date>2022-02-01</date><risdate>2022</risdate><volume>197</volume><issue>2</issue><spage>285</spage><epage>291</epage><pages>285-291</pages><issn>0026-9255</issn><eissn>1436-5081</eissn><abstract>We show that, with the exception of the symmetric derivative, each limit of the form
lim
h
→
0
A
f
(
x
+
a
h
)
+
B
f
(
x
+
b
h
)
h
,
(
A
+
B
=
0
,
A
a
+
B
b
=
1
)
,
is equivalent to the ordinary derivative, for all continuous functions at
x
. And, up to a non-zero scalar multiple, these are the only criteria for differentiating all continuous functions at
x
, by taking limits of first order difference quotients with two function evaluations.</abstract><cop>Vienna</cop><pub>Springer Vienna</pub><doi>10.1007/s00605-021-01574-0</doi><tpages>7</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0026-9255 |
| ispartof | Monatshefte für Mathematik, 2022-02, Vol.197 (2), p.285-291 |
| issn | 0026-9255 1436-5081 |
| language | eng |
| recordid | cdi_proquest_journals_2631890705 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Continuity (mathematics) Derivatives Mathematics Mathematics and Statistics Quotients Tornadoes |
| title | A differentiability criterion for continuous functions |
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