Continuity Integration and Fourier Theory The first part in this thorough textbook is devoted to continuity properties culminating in the theorems of Korovikin and Stone Weierstrass. The last part consists of extensions and applications of the Fourier theory for example the Wilbraham Gibbs phenomenon the Hausdorff Young theorem the Poisson sum formula and the heat and wave equations. Since the Lebesgue integral is indispensible for obtaining familiarity with Fourier series and Fourier transforms on a somewhat higher level the book contains a brief survey with complete proofs of abstract integration theory. The compact and comprehensive exposition is rounded off by well choosen exercises. The book is of interest to advanced undergraduate and graduate students. This book is a textbook on continuity properties integration theory and Fourier theory for graduate or advanced undergraduate students in mathematics or mathematical physics. The discussion of abstract in integration is brief but with complete proofs