The objective of this research paper is to follow up on the work already started in order to install a new mathematical
analysis, the one we called Cartesian analysis see our previous works,. During these last two papers, we started with the
definition of cartesian geometry and we defined and introduced to a new Cartesian topology which gave birth to new spaces
called Saidou spaces. Thus, and to follow up on this way, we propose to studie the analytical and functional part of this analysis.
We will define the notion of a Cartesian function and then its epigraph before to characterize the analytical properties of these
functions, for example the continuity and the differentiablity. In an other hand, we will see the relationship between cartesian
functions and the convex functions. According to the latest papers and this work we do the asset of the first foundations of the