There are also two chapters at the end that develop the topology of the real line and show the equivalence of the non-standard definitions with the standard definitions (and hence the equivalence of the two approaches to calculus). Most the rest of the book develops the standard theorems of single-variable calculus based on hyperreal numbers. This moves along quickly, but is mostly about mathematical logic rather than numbers or calculus, and it’s easy to lose sight of where we want to go with this. Roughly the first third of the book covers the development of the hyperreal number system and the equivalence of logical statements about real numbers and functions with the formally-identical statements about hyperreal numbers and functions. The hyperreal number system is a extension of the reals that embeds them in a very careful way so that most of their properties carry over, but such that there are also infinitesimal and infinite quantities. This is a Dover 2003 unaltered reprint of the 1979 MIT Press edition. The depth of coverage is roughly that of a rigorous introductory analysis book, such as Ross’s Elementary Analysis. It covers all the common theorems of calculus, but does not cover any calculus techniques or applications and has only a modest number of examples. This is a concise, clear, and mostly complete textbook on the theory of calculus, as developed through the ideas of non-standard analysis rather than the epsilon-delta approach.
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