Representation of a Standard Continuous Function by a Microscope
The aim of this paper is to provide a representation of a standard continuous function and a standard differentiable function by mean of a microscope. More precisely, under certain conditions, the following results have been obtained. Let 12F"> be a standard continuous function...
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Mosul University
2010-12-01
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| Series: | Al-Rafidain Journal of Computer Sciences and Mathematics |
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| Online Access: | https://csmj.mosuljournals.com/article_163901_f3f5d9b2f69a490ccc220372eee28a97.pdf |
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| author | Tahir Ismail Hind Saleh |
| author_facet | Tahir Ismail Hind Saleh |
| author_sort | Tahir Ismail |
| collection | DOAJ |
| description | The aim of this paper is to provide a representation of a standard continuous function and a standard differentiable function by mean of a microscope.
More precisely, under certain conditions, the following results have been obtained.
Let 12F"> be a standard continuous function define on 12R"> , and 12°G"> the shadow of it's graph. If there exists a standard point 12X0∈R"> and an interval 12I0"> about 12X0"> such that : 12∀X∈I0,X,FX limited ⟹X≃X0"> .
(i) Furthermore If there exist 12X1"> , 12X2"> limited in 12I0"> such that 12FX1"> , 12FX2"> are infinitely large with opposite sign, then 12°G"> contains the vertical line 12∆"> of the equation 12°X=X0"> .
(ii) If there exist a standard number 12α"> , 12X∈I0"> and if 12FX"> is limited such that 12°FX≤α"> (resp. 12 °FX≥α"> ). Also if there exist 12X1"> , 12 X2"> limited in 12I0"> such that 12FX1<0"> is infinitely large (resp. 12 FX1>0"> ) and 12FX2≃α"> ,then 12°G"> contains the half line 12∆α"> defined by :
12∆α=X,Y∈R2:°X=X0 , °Y≤α resp.°Y≥α ">
Let 12f"> be a standard function defined at a neighborhood at a standard point 12x0"> , then 12f"> is differentiable at 12x0"> <strong>if and only if</strong> under every microscope of power 12ε"> ,centered at 12x0,fx0"> ,the representation of 12f"> is not a vertical line at 12x0,fx0"> . |
| format | Article |
| id | doaj-art-5457f7435beb422f81033ff9a1fb0d2c |
| institution | OA Journals |
| issn | 1815-4816 2311-7990 |
| language | English |
| publishDate | 2010-12-01 |
| publisher | Mosul University |
| record_format | Article |
| series | Al-Rafidain Journal of Computer Sciences and Mathematics |
| spelling | doaj-art-5457f7435beb422f81033ff9a1fb0d2c2025-08-20T02:05:46ZengMosul UniversityAl-Rafidain Journal of Computer Sciences and Mathematics1815-48162311-79902010-12-017211512410.33899/csmj.2010.163901163901Representation of a Standard Continuous Function by a MicroscopeTahir Ismail0Hind Saleh1College of Computer Sciences and Mathematics University of MosulCollege of Computer Sciences and Mathematics University of Mosul, IraqThe aim of this paper is to provide a representation of a standard continuous function and a standard differentiable function by mean of a microscope. More precisely, under certain conditions, the following results have been obtained. Let 12F"> be a standard continuous function define on 12R"> , and 12°G"> the shadow of it's graph. If there exists a standard point 12X0∈R"> and an interval 12I0"> about 12X0"> such that : 12∀X∈I0,X,FX limited ⟹X≃X0"> . (i) Furthermore If there exist 12X1"> , 12X2"> limited in 12I0"> such that 12FX1"> , 12FX2"> are infinitely large with opposite sign, then 12°G"> contains the vertical line 12∆"> of the equation 12°X=X0"> . (ii) If there exist a standard number 12α"> , 12X∈I0"> and if 12FX"> is limited such that 12°FX≤α"> (resp. 12 °FX≥α"> ). Also if there exist 12X1"> , 12 X2"> limited in 12I0"> such that 12FX1<0"> is infinitely large (resp. 12 FX1>0"> ) and 12FX2≃α"> ,then 12°G"> contains the half line 12∆α"> defined by : 12∆α=X,Y∈R2:°X=X0 , °Y≤α resp.°Y≥α "> Let 12f"> be a standard function defined at a neighborhood at a standard point 12x0"> , then 12f"> is differentiable at 12x0"> <strong>if and only if</strong> under every microscope of power 12ε"> ,centered at 12x0,fx0"> ,the representation of 12f"> is not a vertical line at 12x0,fx0"> .https://csmj.mosuljournals.com/article_163901_f3f5d9b2f69a490ccc220372eee28a97.pdffunctionstandardcontinuousdifferentiablemicroscope |
| spellingShingle | Tahir Ismail Hind Saleh Representation of a Standard Continuous Function by a Microscope Al-Rafidain Journal of Computer Sciences and Mathematics function standard continuous differentiable microscope |
| title | Representation of a Standard Continuous Function by a Microscope |
| title_full | Representation of a Standard Continuous Function by a Microscope |
| title_fullStr | Representation of a Standard Continuous Function by a Microscope |
| title_full_unstemmed | Representation of a Standard Continuous Function by a Microscope |
| title_short | Representation of a Standard Continuous Function by a Microscope |
| title_sort | representation of a standard continuous function by a microscope |
| topic | function standard continuous differentiable microscope |
| url | https://csmj.mosuljournals.com/article_163901_f3f5d9b2f69a490ccc220372eee28a97.pdf |
| work_keys_str_mv | AT tahirismail representationofastandardcontinuousfunctionbyamicroscope AT hindsaleh representationofastandardcontinuousfunctionbyamicroscope |