Representation of a Standard Continuous Function by a Microscope
AL-Rafidain Journal of Computer Sciences and Mathematics,
2010, Volume 7, Issue 2, Pages 115-124
AbstractThe 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 12Â°G"> the shadow of it's graph. If there exists a standard point 12X0âˆˆR"> and an interval 12I0"> about 12X0"> such that : 12âˆ€XâˆˆI0,X,FX limited âŸ¹Xâ‰ƒX0"> .
(i) Furthermore If there exist 12X1"> , 12X2"> limited in 12I0"> such that 12FX1"> , 12FX2"> are infinitely large with opposite sign, then 12Â°G"> contains the vertical line 12âˆ†"> of the equation 12Â°X=X0"> .
(ii) If there exist a standard number 12Î±"> , 12XâˆˆI0"> and if 12FX"> is limited such that 12Â°FXâ‰¤Î±"> (resp. 12 Â°FXâ‰¥Î±"> ). Also if there exist 12X1"> , 12 X2"> limited in 12I0"> such that 12FX1<0"> is infinitely large (resp. 12 FX1>0"> ) and 12FX2â‰ƒÎ±"> ,then 12Â°G"> contains the half line 12âˆ†Î±"> defined by :
12âˆ†Î±=X,YâˆˆR2:Â°X=X0 , Â°Yâ‰¤Î± resp.Â°Yâ‰¥Î± ">
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 12Îµ"> ,centered at 12x0,fx0"> ,the representation of 12f"> is not a vertical line at 12x0,fx0"> .
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