I am a college sophomore with a major in mathematics and an aspiring mathematician in the fields of computation theory and cryptography. I am always curious about the importance of "transition books" from calculus to analysis, such as Spivak, Apostol, and Courant. Do I need to read one of them before jumping into real analysis? I took Calculus I-II course using Serge Lang's A First Course in Calculus, and I am currently studying Artin's Algebra and Hoffman/Kunze's linear Algebra, and preparing myself for upcoming research in the computation theory. I will be taking the theoretical linear algebra and abstract algebra for this academic year, and I will proceed to the analysis on next academic year. However, I want to learn more about calculus since I am preparing for Putnam, and measure/approximation theory will be used on my research. Perhaps it is naturally for me to pick up analysis books of Rudin, Apostol, or Pugh, but it seems that quite many people recommend the advanced calculus books of Apostol, Rudin, and Pugh. Is it necessary for me to read those transition books? Can I just jump right into the analysis books? What are the benefits of reading the transition books first before studying the analysis?