Robert Hansen says: >That "intuition" you speak of is not common sense, otherwise we would see it everywhere. I don't see a shortage of common sense in the world, so why is mathematics so difficult for most? The "intuition" you speak of is the underpinnings of analysis and formal reasoning.
Well, I didn't want to get hung up on this point, but its OK. In the case on "continuity" there is a common sense meaning, that is wide-spread enough - it means no gaps, breaks, sudden jumps. A refined mathematical intuition, on the other, starts there and reasons to further no-so-common or easy to understand observations, correlations, conclusions. All I'm saying is much can be done without all the various formal definitions from analysis or topology. Euler and Gauss did OK.
But the starting point is pretty much the same for everyone. And Clyde's version was pretty far from the starting point.
Here's a question - how do you get a student to appreciate that a kind of "reversal" is needed in viewpoint to get to a satisfactory definition. What I'm talking about is the common sense viewpoint takes continuity for granted, and merely looks for "gaps" or "breaks" in the otherwise continuous (think of basic functions like 1/x). The formal definitions though, talk about continuity at a point. In fact one can define a function that is *only* continuous at a single point -- a kind of oxymoron.