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"> .
