On Nov 10, 2012, at 3:26 PM, Joe Niederberger <email@example.com> wrote:
> I would say the common sense view is that a continuous function is one whose graph can be drawn without lifting the pencil off the page.
This is something that I have studied in great detail. It started with the pictorial representations of mathematical concepts, such as Clyde's visualization of the concept of local linearity (zooming in on a function).
There is no such thing as a "common sense view of continuity". That is like saying that the common sense view of gravity is that things fall down. These are not "views" in the sense that a "view" is an understanding or an interpretation of the phenomena. Common sense isn't based on interpretation at all, just observation and acceptance. A common sense "example" of continuity is a function whose graph can be traced without lifting the pencil off the page. But that isn't a big deal is it? The world is chock full of common sense examples of continuity, and of gravity.
What you are trying to say is that a function whose graph can be traced without lifting the pencil is a common sense "interpretation" of continuity. But by itself it is not an interpretation of anything. It is only a damn good interpretation to you and I because we already understand continuity. To a student that is just starting the journey, it is nothing but an example of what we are about to delve into.
All of us tend to confuse what actually occurs when we show a student a concrete example of a mathematical concept. There are two reasons for this. First and foremost is that we have already been there and done that. We understand continuity and we can interpret continuity in examples of continuity. We can be the ball. Secondly, when we think up a damn good example of continuity for our students, we heavily employ our understanding and interpretation of continuity, so much so that we fool ourselves into thinking that the example we just created is more than just an example of continuity but an actual understanding of continuity. All it really is is an example of a teacher that understands continuity well enough to come up with a damn good example of it. It doesn't magically convey that understanding onto the student. For the student, it is nothing but an example. Some examples are better than others, but it isn't understanding. That will require the development of a theory of continuity and that takes time. Hopefully, later, the student can say "Been there, done that!"