In 1894, Stieltjes, while researching the advancements in the field of continued fractions, defined a new method of integration of continuous functions. It is important to fully understand the originality of Stieltjes’ generalisation and how it fundamentally differs from what we have examined so far. In Chap.  1 , we recalled what is referred to as integration in the introductory course of infinitesimal calculus. it is a well-defined operation, which associates a number to each continuous function f(x). In Chaps.  2 , 3 , 6 , 7 , 10 we defined this operation for increasingly larger families of functions f(x). We extended the notion of integration deeper into the realm of functions f(x). Stieltjes, in this case, leaves the family of considered functions f(x) invariable. However, for a given function f(x), he defines as many integrals as we want. Each of them associates a number to f(x). He extends the concept of surface integration into the field of functional operations.