A differentiability criterion for continuous functions
Abstract We show that, with the exception of the symmetric derivative, each limit of the form limh→0Af(x+ah)+Bf(x+bh)h,(A+B=0,Aa+Bb=1),$$\begin{aligned} \lim _{h\rightarrow 0}\frac{Af(x+ah)+Bf(x+bh)}{h},\qquad (A+B=0,Aa+Bb=1), \end{aligned}$$is equivalent to the ordinary derivative, for all continuo...
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| Autor*in: |
Catoiu, Stefan [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2021 |
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| Anmerkung: |
© The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2021 |
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| Übergeordnetes Werk: |
Enthalten in: Monatshefte für Mathematik - Springer Vienna, 1948, 197(2021), 2 vom: 24. Mai, Seite 285-291 |
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| Übergeordnetes Werk: |
volume:197 ; year:2021 ; number:2 ; day:24 ; month:05 ; pages:285-291 |
| Links: |
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| DOI / URN: |
10.1007/s00605-021-01574-0 |
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OLC2078105392 |
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| 520 | |a Abstract We show that, with the exception of the symmetric derivative, each limit of the form limh→0Af(x+ah)+Bf(x+bh)h,(A+B=0,Aa+Bb=1),$$\begin{aligned} \lim _{h\rightarrow 0}\frac{Af(x+ah)+Bf(x+bh)}{h},\qquad (A+B=0,Aa+Bb=1), \end{aligned}$$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. | ||
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10.1007/s00605-021-01574-0 doi (DE-627)OLC2078105392 (DE-He213)s00605-021-01574-0-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 7170 VZ rvk SA 7170 VZ rvk Catoiu, Stefan verfasserin aut A differentiability criterion for continuous functions 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2021 Abstract We show that, with the exception of the symmetric derivative, each limit of the form limh→0Af(x+ah)+Bf(x+bh)h,(A+B=0,Aa+Bb=1),$$\begin{aligned} \lim _{h\rightarrow 0}\frac{Af(x+ah)+Bf(x+bh)}{h},\qquad (A+B=0,Aa+Bb=1), \end{aligned}$$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. Generalized Riemann derivative -derivative Peano derivative Enthalten in Monatshefte für Mathematik Springer Vienna, 1948 197(2021), 2 vom: 24. Mai, Seite 285-291 (DE-627)129492191 (DE-600)206474-1 (DE-576)014887657 0026-9255 nnns volume:197 year:2021 number:2 day:24 month:05 pages:285-291 https://doi.org/10.1007/s00605-021-01574-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_267 GBV_ILN_2088 GBV_ILN_4012 SA 7170 SA 7170 AR 197 2021 2 24 05 285-291 |
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10.1007/s00605-021-01574-0 doi (DE-627)OLC2078105392 (DE-He213)s00605-021-01574-0-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 7170 VZ rvk SA 7170 VZ rvk Catoiu, Stefan verfasserin aut A differentiability criterion for continuous functions 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2021 Abstract We show that, with the exception of the symmetric derivative, each limit of the form limh→0Af(x+ah)+Bf(x+bh)h,(A+B=0,Aa+Bb=1),$$\begin{aligned} \lim _{h\rightarrow 0}\frac{Af(x+ah)+Bf(x+bh)}{h},\qquad (A+B=0,Aa+Bb=1), \end{aligned}$$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. Generalized Riemann derivative -derivative Peano derivative Enthalten in Monatshefte für Mathematik Springer Vienna, 1948 197(2021), 2 vom: 24. Mai, Seite 285-291 (DE-627)129492191 (DE-600)206474-1 (DE-576)014887657 0026-9255 nnns volume:197 year:2021 number:2 day:24 month:05 pages:285-291 https://doi.org/10.1007/s00605-021-01574-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_267 GBV_ILN_2088 GBV_ILN_4012 SA 7170 SA 7170 AR 197 2021 2 24 05 285-291 |
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Abstract We show that, with the exception of the symmetric derivative, each limit of the form limh→0Af(x+ah)+Bf(x+bh)h,(A+B=0,Aa+Bb=1),$$\begin{aligned} \lim _{h\rightarrow 0}\frac{Af(x+ah)+Bf(x+bh)}{h},\qquad (A+B=0,Aa+Bb=1), \end{aligned}$$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. © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2021 |
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Abstract We show that, with the exception of the symmetric derivative, each limit of the form limh→0Af(x+ah)+Bf(x+bh)h,(A+B=0,Aa+Bb=1),$$\begin{aligned} \lim _{h\rightarrow 0}\frac{Af(x+ah)+Bf(x+bh)}{h},\qquad (A+B=0,Aa+Bb=1), \end{aligned}$$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. © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2021 |
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Abstract We show that, with the exception of the symmetric derivative, each limit of the form limh→0Af(x+ah)+Bf(x+bh)h,(A+B=0,Aa+Bb=1),$$\begin{aligned} \lim _{h\rightarrow 0}\frac{Af(x+ah)+Bf(x+bh)}{h},\qquad (A+B=0,Aa+Bb=1), \end{aligned}$$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. © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2021 |
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2024-07-03T18:50:37.661Z |
| _version_ |
1803584943070117888 |
| fullrecord_marcxml |
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