There seems to be much confusion and quibbling about the meaning of "common sense" ...
I certainly didn't mean to quibble about the meaning of common sense, contrast it with a keen developed intuition, or even imply what is common for one is always the same for all.
I agree that most algebra/pre-calc students if willing and led by a competent teacher can appreciate the mathematical concept of continuity. I also realize now that was your position - you weren't saying the picture you presented was a staring place, it was the arrival place.
Let's see the starting place then.
I'll just add a few observations. I'm not pretending to speak as an expert educator - but this is just my view on this.
* The mathematical concept certainly does arise from our common experience.
Just the other day my son was asking why a line is continuous.:)
I have never heard of anyone looking at a graph of a function and saying "Hey, look! That is continuous." Mathematicians asked about continuity because in their investigations into other problems (a lot of other problems) they started to realize that some notion of continuity was involved. I am asking about common sense now because in reviewing dozens of curriculums, exams and teaching methods, it dawned on me that I could classify ALL of them by their use and treatment of common sense. I realized that there was a line between common sense and reasoning and they were not connected. Some curriculums understood this, some did not. I realized overall that our common sense notion of common sense is flawed.
You cannot ground the mathematical notion of anything in common sense. That is like saying we grounded Einstein's special theory of relativity in the observation that Mercury was a fraction of a second late in its orbit. You ground mathematical notions in other mathematical notions, and we call all of that a theory. It is no easy task and it requires uncommon sense. It requires formal reasoning. We use common sense examples as an aid in talking to that theory so that the student isn't in the position of building a ship in a bottle, blindfolded, and never haven seen a ship before, or a bottle. Eventually though, as we get deeper into mathematics, those aids become much more sparse.
I have studied the curriculums. When they attempt to ground math in common sense, they fail. How do I know they fail. Because they never stop. If you show me a book on playing the piano and in chapter one you are using a technique where you hold the student's hands and lightly press on the fingers they are supposed to play, I might say "That's interesting, does it work?" If by chapter 12, you are still doing this, then I have my answer.
The fact that we are even having this conversation proves that math is not grounded in common sense. Everyone has common sense. That is why they call it "common" sense. Obviously, this is not a sufficient requirement for mathematics. Q.E.D.