TITLE: What is the theme of analysis?
QUESTION [55 upvotes]: It is safe to say that every mathematician, at some point in their career, has had some form of exposure to analysis. Quite often, it appears first in the form of an undergraduate course in real analysis. It is there that one is often exposed to a rigorous viewpoint to the techniques of calculus that one is already familiar with. At this stage, one might argue that real analysis is the study of real numbers, but is it? A big chunk of it involves algebraic properties, and as such lies in the realm of algebra. It is the order properties, though, that do have a sort of analysis point of view. Sure, some of these aspects generalise to the level of topologies, but not all. Completeness, for one, is clearly something that is central to analysis.
Similar arguments can be made for complex analysis and functional analysis.
Now, the question is: As for all the topics that are bunched together as  analysis, is there any central theme to them? What topics would you say that belongs to this theme? And what are the underlying themes in these individual subtopics?
Add. It may be a subjective question, but having a rough idea of what the central themes of a certain field are helps one to construct appropriate questions. As such, I think it is important. I am not expecting a single answer, but more of a diverse set of opinions on the matter.

REPLY [8 votes]: One additional central aspect of analysis are approximations. After all, the very limit notation is nothing but a qualitative approximation.
In its core it means "The more effort you put into your approximation, the more precise it will be".
For example, continuity gives you the guarantee, that even if you miss the true input value of the function, your output value will still be "close". Just imagine if reading a number of a ruler wasn't continuous - there'd be hardly any point in making the measurement in the first place!
Differentiability gives you an approximation of an interval, in which the function is monotone.
Integration gives you an approximation of the mistake you make, when you approximate a function into segments.
In many cases, analysis gives us results that the difference of an approximation to the original object goes to 0 in the limit.
If we look at this from the practical point of view, where everything has an error term, this is all the justification we need to swap out the original with the approximation.
(Yes, if you have a calculation that isn't e.g. continuous, you'll get rubbish - but as any representation of the original object would have an error, you'd always anyway get rubbish)