I How can we characterize mathematical analysis?

[Mr Davis 97]

If algebra can be characterized as the study of algebraic structures and the structure-preserving maps between them, how could characterize mathematical analysis? In analysis, can we break it down into certain mathematical structures, maybe topologies or metric spaces, and the continuous maps between them? Would this be the best way to characterize analysis?

 

[fresh_42]

I'd say it's about functions $f\, : \,\mathbb{K}^n \longrightarrow \mathbb{K}^m$ where $\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}$ all equipped with the Euclidean metric and about these spaces themselves.

The first part includes various continuity concepts as well as all differentiabilty degrees, and the second part topological features like sequences, series and infinite products and measure theory.

I can't imagine a specific categorial classification as topological spaces, vector spaces or measure algebras would be as it is all of them restricted to $\mathbb{R}$ or $\mathbb{C}$ as basic field, or at least to fields of characteristic $0$. Maybe the most general description would be: The Theory of the Equations of Movement.

 

[FactChecker]

The "structure-preserving maps" part is too specific to abstract algebra. A lot of mathematical analysis is about other properties.