Why weren't continuous functions defined as Darboux functions? 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?

The Darboux definition does not correspond very well with our intuition about continuity.  For example, the Conway function takes on every value in every interval, and is therefore Darboux.  However it is not continuous, and I don't think we want it to be continuous, because it certainly doesn't agree with your teacher's definition of drawing without lifting your pencil.
