> While teaching removable and non-removable discontinuities > today I had a student ask me what the purpose is and why > it is useful to be able to distinguish between them...off > the top of my head I didn't have a great answer...is it > just the value of analyzing a function or is there a grander > way that it fits into the picture of calculus that I'm not > thinking of?
You could point out that a function with a (an isolated) removable discontinuity can be made continuous (at least, continuous in a neighborhood of the point) by changing the value of the function at a single point, but in the case of a non-removable discontinuity you have to change the value of the function at infinitely many points. Thus, a removable discontinuity is rather innocuous, being a situation in which the function was "accidentally given" the wrong value (or no value) at the point.
In 1800s textbooks the phrase "true value" was often used for the limiting value of a function at a removable discontinuity point. Also, "vanishing fraction" was often used for 0/0 indeterminate forms. The following google-books search, which simultaneously searches for both these phrases, leads to a lot of freely available 1800s literature where these terms came up, and looking at some of these could be something some readers here (or their students) might find interesting.