>What I said was that understanding the meaning of drawing a curve without lifting your pencil is NOT the same as understanding the meaning and significance of continuity.
You keep talking about "the significance of X" -- last time it was variables. So, in a nutshell, what is the "meaning and significance" of continuity? How does it differ from an understanding of continuity?
I tossed out the old pencil bit as an example of a starting point for discussion, not as the be all and end all, so I don't get how your supposed observation even applies to what I said. Nor do I understand your "small engine bit" -- I can only guess you think starting with Weierstrass epsilon-delta definition is the *only* way to go. Nonsense. Nor is it the be all and end all definition of continuous function.
I was hoping the notion of focusing rather on discontinuities (at first) would resonate with someone, at least. I guess I'm that old fashioned. The oscillating type is a mind bender for young folk. I would make connections with other scale invariant structures.