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Topic: Is (t^2-9)/(t-3) defined at t=3?
Replies: 166   Last Post: Oct 30, 2013 9:41 AM

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quasi

Posts: 9,923
Registered: 7/15/05
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Oct 8, 2013 11:53 PM
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I've now had the time to do a little checking.

Some of the books I checked avoided the issue entirely by never
using the phrase "f is continuous" without qualifying it by
saying either

"f is continuous at [some point]"

"f is continuous on [some set]"

Examples of such books are:

Larson & Edwards
Calculus, 9th Ed (2010)

Spivak
Calculus, 3rd Ed (1994)

Stewart
Calculus - Early Transcendentals, 5th Ed (2002)

On the other hand, it seems Arturo Magidin wasn't wrong.
Some books say it exactly the way he did. Examples of such
books are:

Adams
Calculus - A Complete Course (2006)

Blank & Krantz
Calculus - Single Variable, 2nd Ed (2011)

Thomas & others
Calculus, 11th Ed (2004)

Their version of "f is continuous" yields the advantage of being
consistent with definition of continuous function used in later
courses such as Topology, traded off against the disadvantage of
being perhaps inconsistent, in some cases, with the earlier
notion of continuous function introduced in Precalculus.

But since at least _some_ books match Arturo's version perfectly,
and _none_ match the definition I proposed, I withdraw my
proposed definition.

Still, all in all, I think it makes sense, at the Elementary
Calculus level, to avoid potential confusion by never just saying
"f is continuous", but rather always qualifying it using one of
the forms

"f is continuous [at some point]"

"f is continuous [on some set]"

except in cases where omitting the qualification would
not contradict earlier Precalculus notions.

quasi


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9/28/13
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