TITLE: Why weren't continuous functions defined as Darboux functions?
QUESTION [38 upvotes]: When we were in primary school, teachers showed us graphs of 'continuous' functions and said something like 

"Continuous functions are those you can draw without lifting your pen"

With this in mind I remember thinking (something along the lines of)
"Oh, that must mean that if the function takes two values $f(y)<f(z)$ then 
for every $c$ between $f(y), f(z)$ there must be some $x\, (y<x<z)$ such that $f(x)=c$"
And that's what I thought a continuous function was. But then the $\epsilon$-$\delta$ definition appeared, which put a more restrictive condition on what a continuous function was.
So, my question is, given the fact that Darboux functions "seem continuous" (in some subjective sense, I guess), why wasn't this used as the definition of continuity? More generally, how did today's (analytical) definition of continuity appear?

REPLY [4 votes]: I want to focus on your question "why wasn't this used as the definition of continuity?" In other words, you're asking why we don't interpret continuity to mean "having the intermediate value property."
One answer that hasn't been mentioned yet is that continuity can be a purely local phenomenon. That is, it makes sense to ask whether a function is continuous or discontinuous at a point. By contrast, one cannot ask whether a function possesses the intermediate value property at a point. My understanding is that in the early history of analysis, the local and global aspects of continuity often weren't carefully distinguished -- or rather, they weren't even recognized as different. This is why, for example, many could confuse continuity and uniform continuity. The latter is a strictly global phenomenon, too: it does not make sense to ask whether a function is uniformly continuous at a point. But as it became clear that continuity could be interpreted as a pointwise phenomenon, these different shades of regularity -- local versus global -- came into view.
Added:
Of course, since I mentioned shades of regularity, I might as well point out that the intermediate value property tells you nothing about smoothness (differentiability). If you want continuity to be the first step in a hierarchy of degrees of differentiability, you need to look for a different notion of continuity than the IVP. I don't know exactly how the history played out, but you can see why, as it became clear that differentiability too is a pointwise phenomenon, one would have looked for a notion of pointwise continuity to fit into this framework. (Compare this to the search for the right notion of differentiability for multivariable functions: you want to pick out the right definition to guarantee continuity.)