R. L. Goodstein 
Recursive Analysis [EPUB ebook] 

Recursive analysis develops natural number computations into a framework appropriate for real numbers. This text is based upon primary recursive arithmetic and presents a unique combination of classical analysis and intuitional analysis. Written by a master in the field, it is suitable for graduate students of mathematics and computer science and can be read without a detailed knowledge of recursive arithmetic.Introductory chapters on recursive convergence and recursive and relative continuity are succeeded by explorations of recursive and relative differentiability, the relative integral, and the elementary functions. A final chapter examines transfinite ordinals, and the text concludes with a helpful appendix of topics related to recursive irrationality and transcendence.