Representation of standard Liebchtz functions under a microscope

ABSTRACT The aim of this paper is to provide a representation of a standard lipschitzian functions by mean of a microscope. More precisely, under certain conditions, the following results have been obtained. Let be a standard function, and be the shadow of its graph : (i) If is s-continuous and limited, then is the graph of a continuous function on . (ii) If is continuous and if is the graph of a function defined on , then is a continuous function. The standard function is lipschitzian at a neighbourhood of a standard point if and only if is limited under every microscope of power centered at a point or at a point infinitely close to it. If the standard function is lipschitzian at a neighbourhood of a standard point ,then it's representation under a microscopic of power centered at the point or at a point infinitely close to it, is a graph of a lipschitzian function.