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Topic: [ap-calculus] Discontinuous at this point because .....
Replies: 2   Last Post: Sep 12, 2012 4:05 PM

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Dave L. Renfro

Posts: 2,165
Registered: 11/18/05
Re:[ap-calculus] Discontinuous at this point because .....
Posted: Sep 12, 2012 4:05 PM
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Kristin Visser wrote:

http://mathforum.org/kb/message.jspa?messageID=7888724

> 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.

http://www.google.com/search?q=%22true%20value%22%20%22vanishing%20fraction%22&tbm=bks

Dave L. Renfro
====
Course related websites:
http://apcentral.collegeboard.com/calculusab
http://apcentral.collegeboard.com/calculusbc
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