Limit of a Function and Programming

I just finished teaching two Mathematical Analysis classes here at Chilpancingo. As a physicist I rarely had the chance to teach these classes, usually it is the math types that have that chore. I am glad I did.

My official concern at this point in my life is to find new ways to teach mathematics. I had a hard time with my first calculus class as a freshman in college. In high school I had mastered the calculus algorithms, without concern with rigor. When Cauchy entered in my life I received him with some kind of confusion. What was the point of all that rigmorole of epsilons and deltas. Eventually I got it, otherwise I wouldn't have gotten as far as I did with my science education.

It was only now that I was teaching these graduate classes that it hit me. I did not understand because nobody told me that the definition of continuity is not a formula, it is a program.

Stephen Wolfram in a New Kind of Science clearly states that nature's mathematical language is not expressed in formulas, but programs. Readers that do not know calculus or programming, likely do not understand what I mean. I make an effort here to put this idea in lay language.

A function is continuous if you can draw a graph of it without lifting the pencil from the paper. A function is a relation between  numbers, some come in and other comes out. You want to know the area of a room a feet long and b feet wide, say. The answer is, multiply a and b.  The input here are a, and b, the output a times b. Now imagine that you don't know what a is, so you draw this product for several values of a. When you get the   total area you want, say 10 squared feet, then you stop and look up the value of a. When doing this drawing you do not have to lift the pencil, it continuously stays on the graph.

Even though I described a process where the value of a changed, the answer to the problem posed is a formula. The continuity definition I struggled with when I was a freshman goes something like this: You have two perpendicular lines, one is a, the other a times b, with b equals 1 foot ,say. Now Cauchy taught us that the function is continuous, if and only if, we take a set of values in the a times b vertical line, and consider those in the horizontal line; no matter how small the region around one point, one must have a small region on both lines. If one makes the vertical line region smaller, the horizontal line region will get also smaller, in a never ending story.

Cauchy's definition is a program, and nobody told me so!

Maybe if we tell the students, they will understand it better.