Reception
The book was first reviewed by Errett Bishop, noted for his work in constructive mathematics. Bishop's review was harshly critical and has been described as vitriolic by historian Joseph Dauben. His review caused a controversy, which is detailed at the article Criticism of non-standard analysis. Shortly after Martin Davis and Hausner published a detailed favorable review, as did Andreas Blass and Keith Stroyan. Keisler's student K. Sullivan, as part of her Ph.D. thesis, performed a controlled experiment involving 5 schools which found Elementary Calculus to have advantages over the standard method of teaching calculus. Despite the benefits described by Sullivan, the vast majority of mathematicians were not convinced to adopt infinitesimal methods in their teaching. Recently, Katz & Katz give a positive account of a calculus course based on Keisler's book. O'Donovan also described his experience teaching calculus using infinitesimals. His initial point of view was positive, but later he found pedagogical difficulties with approach to non-standard calculus taken by this text and others..
Hrbacek writes that the definitions of definitions of continuity, derivative, and integral implicitly must be grounded in the ε-δ method in Robinson's theoretical framework, in order to extend definitions to include non-standard values of the inputs, claiming that the hope that non-standard calculus could be done without ε-δ methods could not be realized in full. Błaszczyk et al. detail the usefulness of microcontinuity in developing a transparent definition of uniform continuity, and characterize Hrbacek's criticism as a "dubious lament".
Read more about this topic: Elementary Calculus: An Infinitesimal Approach
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