Step 1: Learn the basic knowledge of set theory. Functional basis. High school textbooks have these contents.
Step 2: Learn Mathematical Analysis. The related self-study textbook: Mathematical Analysis (Volume I), edited by Fudan University Press, Ouyang Guangzhong, Yao Yunlong and Joo Won?. This is a good textbook. In order to learn the theory of real variable function, it is enough for you to learn the first book by yourself. As for the second volume, it is the content of multivariate functions. This book is well proved, easy to understand, thorough in explanation and detailed in answers to exercises.
The third part: With the knowledge of mathematical analysis, I believe you will easily learn the theory of real variable function. There are many textbooks about the theory of real variable function, the most famous of which is Tang Song's Russian textbook "The Theory of Real Variable Function" (5th edition). But one disadvantage is that the exercises are difficult and there are no answers in the book. But you can download the answers online. Sina likes these things very much. There are also many excellent textbooks on real variable function theory. If you really want to teach yourself the theory of real variable function, I suggest you buy more textbooks about it, so that you can not be bound by one or two.
These textbooks are generally not available in bookstores. If the landlord is a college student, you can go to the university library, where there should be some. Alternatively, you can buy these books online. Dangdang, Taobao and Joyo Amazon all have them.
Because some knowledge points of real variable function theory are difficult, if you encounter problems that cannot be solved temporarily, I suggest you put them aside first, or ask others. It is best to choose famous textbooks, because famous books have their own reasons, which are generally better, well organized and suitable for self-study.
I am also a self-taught person. At present, I teach myself linear algebra. I am a kindred spirit, with support!