الملخص

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">  .