Pathways to Real Analysis provides an introduction to several key ideas of real analysis, from Archimedes quadrature of the parabola, to the Calculus of Newton and Leibniz, power series, Cauchy’s definitions of limit and integral, the inverse function theorem, the implicit function theorem, the wave equation, Fourier’s heat equation and Fourier series. The book provides pathways of discovery that are mathematically natural. Examples are strategically selected in order to help the reader obtain the appropriate insights. Eventually, this initial understanding can be subsumed under a further context where one would explore and establish proofs in an axiomatic context. Prior to proving a result, however, it is helpful to first have discovered a result as a possibility. The main objective of this book is to help promote that initial discovery, especially as relevant to the emergence of real analysis. This is a mathematics book for college students and college teachers in science, technology, engineering and mathematics (STEM) and could serve as a supplement to a calculus sequence such as differential, integral and multi-variable calculus (Calculus I, II and III), or as a textbook for an introduction to real analysis.

Preface / Acknowledgements / Discovering the Inverse Function Theorem / Discovering Calculus / Discovering Real Analysis / Epilogue, Complex Numbers, Complex Analysis and Beyond / Bibliography / Index.

Undergraduate and Postgraduate Students