TITLE: Why do introductory real analysis courses teach bottom up?
QUESTION [31 upvotes]: A big part of introductory real analysis courses is getting intuition for the $\epsilon-\delta\,$ proofs. For example, these types of proofs come up a lot when studying differentiation, continuity, and integration. Only later is the notion of open and closed sets introduced. Why not just introduce continuity in terms of open sets first (E.g. it would be a better visual representation)? It seems that the $\epsilon-\delta$ definition would be more understandable if a student is first exposed to the open set characterization.

REPLY [4 votes]: One reason for the traditional setup (starting with $\varepsilon$ and $\delta$)
may be that not all questions in real analysis can be reduced to topology
alone. Depending on the curriculum, the need to connect the material with
prior exposure to calculus may also play a role.
Although the definition of continuity via open sets is elegant and effective, 
I do not think it to be particularly intuitive for beginners. It also does
not give a good description of continuity at a point. Instead I prefer to
define continuity via neighbourhoods. This approach is used e.g. in the
books of Jameson "Topology and normed spaces" or Brown "Elements of modern
Topology". The advantage is that you can start with the notion of a system
of neighbourhoods of a point (the only condition being that a neighbourhood
$U$ of a point $x$ must satisfy $x\in U$) and state the definitions for a
map to be continuous 
$f: X \to Y$ is continuous at a point $x\in X$ if for every neighbourhood
$U\ni f(x)$ there is a neighbourhood $V\ni x$ with $f(V)\subseteq U$.
$f: X \to Y$ is continuous if it is continuous at every point $x\in X$.
before you go into the precise conditions for a neighbourhood system.
In particular you can take open intervals or open disks in the plane
as examples of neighbourhoods and provide pictures. This can then also
be translated into the $\varepsilon-\delta$ statements, but the intuition
can be built first.