Representation of a Standard Continuous Function by a Microscope
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 12G"> the shadow of it's graph. If there exists a standard point 12X0R"> and an interval 12I0"> about 12X0"> such that : 12XI0,X,FX limited XX0"> .
(i) Furthermore If there exist 12X1"> , 12X2"> limited in 12I0"> such that 12FX1"> , 12FX2"> are infinitely large with opposite sign, then 12G"> contains the vertical line 12"> of the equation 12X=X0"> .
(ii) If there exist a standard number 12"> , 12XI0"> and if 12FX"> is limited such that 12FX"> (resp. 12 FX"> ). Also if there exist 12X1"> , 12 X2"> limited in 12I0"> such that 12FX1<0"> is infinitely large (resp. 12 FX1>0"> ) and 12FX2"> ,then 12G"> contains the half line 12"> defined by :
12=X,YR2:X=X0 , Y resp.Y ">
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 12"> ,centered at 12x0,fx0"> ,the representation of 12f"> is not a vertical line at 12x0,fx0"> .
