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Representation of a Standard Continuous Function by a Microscope | ||
| AL-Rafidain Journal of Computer Sciences and Mathematics | ||
| Article 21, Volume 7, Issue 2, 2010, Pages 115-124 PDF (660.85 K) | ||
| Document Type: Research Paper | ||
| DOI: 10.33899/csmj.2010.163901 | ||
| Authors | ||
| Tahir H. Ismail1; Hind Y. Saleh2 | ||
| 1College of Computer Sciences and Mathematics University of Mosul | ||
| 2College of Computer Sciences and Mathematics University of Mosul, Iraq | ||
| Abstract | ||
| 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"> if and only if under every microscope of power 12ε"> ,centered at 12x0,fx0"> ,the representation of 12f"> is not a vertical line at 12x0,fx0"> . | ||
| Keywords | ||
| function; standard; continuous; differentiable; microscope | ||
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