This book introduces the elements of the theory of real-valued functions of a real variable. The book is aimed at young mathematicians and others who would like to see a coherent account of basic analysis as a rigorous mathematical theory. It aims to avoid various extremes: It does not brush any of the serious difficulties in analysis proper under the carpet. The reader is assumed to be bright, and willing to work hard. It does not dwell on routine stuff. The reader has probably already taken a non-rigorous calculus course, and does not need to be led through that material again.