The word Analysis stems from the bringing together of the words "Isis" (the Egyptian god of fertility and nookie) and "Anal" (meaning "Anal"). This is now believed to be a subtle joke on behalf of the Mathemagicians since Analysis is almost entirely unrelated to anal sex. Some european etimologists relate the orygin of this word to the ass, of course, and the ancient greek "lysis", which means break, fracture. This makes sense in the anal way of life.
Another etymology is based on a decomposition of "analysis" into "An-" "-a-" and "-lysis. Since "lysis", as defined just before, means to break or to fracture, "a-lysis" hence would mean "to not break" and applying the same reasoning An-alysis would thus mean to not not break, ie. to break.
Freshmen entering university for the first time needed something else to call Calculus to show that they were proper, dyed-in-the-wool mathemagicians.
Thus Analysis was developed. It is the study of Calculus, except that the proofs by Hand waving are removed and replaced with long, complicated proofs involving epsilon, or, failing that, anal sex.
The use of Epsilon is central to many proofs in Analysis, and was first defined by Cauchy to be: a very, very small number. No, smaller than that. Like, so small you can hardly think of it at all. Then take half of that. Groovy.
As such, Epsilon is generally quite poorly defined, which tends to ease the difficulty of proving things in Analysis since you can just mutter something about Epsilon and get away with it. This may seem at first sight to be analogous to Hand waving, but since Epsilon is involved it is clearly not. This result, credited to Cauchy, is known as Gödel's Incompleteness Theorem.
Once students have reached sufficient skill in analysis, they are ready to start studying "applied analysis". This involves using analysis to solve real-life problems. As such, it is impractical to go through the long and complicated proofs involving Epsilon, so these are removed from the topic and replaced by Hand waving. But because it's all very complicated mathematics, this is all fine.
Another branch of analysis is "Cauchy Analysis", later renamed "Complex Analysis". Nobody is quite sure what this is about, but before he died Cauchy managed to convince the mathematical world (using a proof known as Cauchy's Bassoon) that it was all very important and definitely worth the price of keeping him on as a professor and feeding him cake.
Examples of deconstructive analysis include Peedu.