This text is an outgrowth of lectures given at the University of Windsor, Canada. One of our main objectives is updating the undergraduate analysis as a rigorous postcalculus course. While such excellent books as Dieudonn´e's Foundations of Modern Analysis are addressed mainly to graduate students, we try to simplify the modern Bourbaki approach to make it accessible to sufficiently advanced undergraduates.

On the other hand, we endeavor not to lose contact with classical texts, still widely in use. Thus, unlike Dieudonn´e, we retain the classical notion of a derivative as a number (or vector), not a linear transformation. Linear maps are reserved for later (Volume II) to give a modern version of differentials. Nor do we downgrade the classical mean-value theorems or Riemann–Stieltjes integration, but we treat the latter rigorously in Volume II, inside Lebesgue theory. First, however, we present the modern Bourbaki theory of antidifferentiation, adapted to an undergraduate course.