The first is the concept of "limit", which is "?" You must be studious and be "careful" at first, that is to say, you must strictly follow this definition, so that you can avoid such questions as "why does this need certification" and "why is this so troublesome to prove".

Second: ruin your three views. See more counterexamples: continuous but non-derivable, original function exists but Riemann is not integrable, discontinuous everywhere, continuous but not monotonous everywhere, continuous but not derivable everywhere, derivable but not monotonous everywhere. As long as you know that these deep-well ice-like functions exist, you will "dare not be arbitrary" when you prove them. Welcome to counterexample analysis, it is really a functional mental hospital.

Third: do the questions in moderation, don't brush a few Midovic, the efficiency is too low, you can do some simplified versions, understand first, and then calculate. Don't switch between limit and integral easily, and don't switch between two limits easily. Dare not Taylor expand any function. I think Pei Liwen's Typical Examples in Mathematical Analysis is better, but it is a bit difficult. Novices don't even look at Rudin, and it's boring to play until you die. There is a set of three volumes of Selected Translation of Russian Mathematics Textbooks and Calculus Course (calculus, but rigor is enough), which is relatively unpretentious, suitable for beginners and rich in content. You can omit the parts that you are not interested in when reading. I watched this set when I was a freshman in physics department, and then I went to the math department to watch Rudin's Principles of Mathematical Analysis again. I think Rudin had better learn it again. Also, if you are interested in how to calculate the integral, you can read a book:

Paul J. Nashin in Interesting Integral

Fourth: the topic still needs to be done. Studying math is also afraid of what you think you know. Many high school students in Zhihu claim to have studied mathematical analysis. In order to test yourself, after-school exercises should be done, and at least 80%-90% can be done correctly. Do more comprehension/proof questions and make appropriate calculations. Even if you can't do it, ask someone else. You can't give up quality for the sake of learning speed. The end result is suicide.

reference data

How to learn mathematical analysis well? . Zhihu [citation time 20 18-3-9]