The most difficult branch of mathematics, when I know calculus, I think it is the most difficult, but in front of complex variable function, I think it is the most difficult, and real variable function is defeated by pan-analysis. It's hard to say which branch is difficult. Your so-called failure is only the depth of the textbook. For example, real variable function and functional analysis, if you use Zhou Minqiang's real variable textbook and an ordinary functional textbook, I am afraid you will get the opposite conclusion. What I said later is not accurate either. Tensor, for example, is an important tool in basic disciplines, abstract tensor is the basic concept of commutative algebra, and various concrete tensors provide important geometric structures in differential geometry. As far as comprehensiveness is concerned, algebraic geometry, noncommutative geometry and mathematical physics may be the subjects that use the most branches of various disciplines. Studying them requires a lot of tools of algebraic analysis and geometry, which requires a lot of knowledge accumulation.