The first is the definition and structure of "real number". This basic problem actually involves the underlying concept of "completeness". It is hard to understand that beginners need to go beyond "experience intuition" and move towards rational and axiomatic thinking. From what I saw on Zhihu, many people didn't master it well. Interestingly, many people thought they did. Their understanding is often half experience, half mathematics and half wrong. Some people tend to go in the direction of "philosophy".
Only by mastering real numbers can a series of subsequent Cauchy discriminant methods be truly established. There is not much difficulty in the series. If you understand the nature of real numbers, the difficulty of sequence is the concept of upper and lower limits. Using this concept well can help you solve many problems.
The difficulty of function continuity lies in various equivalent descriptions of continuity. Many people can't grasp the perspective of understanding continuity through "open collection". In fact, this is a very natural perspective. This involves the problem of "topology", which can essentially return to the properties of real numbers, such as interval sets and open covering theorems. The differentiation of functions is actually the mathematicization of the concept of linear approximation, and the difficulty is actually the fancy application of various mean value theorems. This is also a crazy place for many beginners. They don't know how the buildings that fell from the sky came from. In fact, many of them are just experiences. When it comes to Riemann integral, the real difficulty is not how to understand it through the upper and lower branches, but how to deal with the love-hate relationship between "function limit, differential and integral". The first relationship is
The relationship between indefinite integral and Riemann integral, the existence of primitive function, and the existence of Riemann integral? Then there is the most important "fundamental theorem of calculus" between differential and integral. Limit problem of function under integral. Many people give up treatment completely when they arrive, and all kinds of orders are exchanged casually. Treating indefinite integral and definite integral as one thing is the question of whether the sum with c is the upper and lower limit. Many people focus on finding all kinds of "singular integrals", which is completely putting the cart before the horse. In essence, it is because many people regard "mathematical analysis" as a subject that only values calculation. In fact, its most important thing is understanding, which is also the biggest difference from advanced mathematics/calculus.