CALCULUS, by Tom Apostol, develops mathematical analysis based on the notions of number and real straight up to the derivatives in one and several variables and the integrals of both Riemann and Lebesgue, through the concepts of number, succession or boundaries. It also brings an introduction to the complex calculation.
As to what is customary in contemporary books, is based on notions of topology (explained in the second theme of the book) for demonstrations of the various theorems. The advantages of this lies in the simplicity of these shows using the topology tools, the problem is that the student may not present enough slack to follow them a clear way.
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