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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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@less@ndro

Posts: 216
Registered: 12/13/04
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Oct 30, 2013 9:41 AM
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quasi <quasi@null.set> wrote:
> Marc Olschok wrote:
> >quasi wrote:
> >

> >> As I've previously suggested, the wording is key ...
> >>
> >> Consider the following problems ...
> >>
> >> Problem (1)
> >>
> >> Let f(x) = (x^2 - 9)/(x - 3).
> >>
> >> (a) Is f(3) defined?
> >>
> >> answer: No.
> >>
> >> (b) Is f continuous on its domain?
> >>
> >> answer: Yes.
> >>
> >> (c) Is f continuous on the set of real numbers?
> >>
> >> answer: No, f is not continuous at x = 3.

> >
> >I have not read everything in this (IMHO broken) thread, so
> >this might already have been pointed out. The problem is that
> >question (c) and its answer is meaningless, once it is settled
> >that 3 is not in the domain of f. One might as well ask if f
> >is continuous on the quaternions.

>
> Not really.
>
> In the Calculus context, the missing point scenario is
> classified as either a removable discontinuity or a
> non-removable discontinuity depending on whether the
> relevant limit exists.
>
> In particular, the function
>
> f(x) = (x^2 - 9)/(x - 3)
>
> is said to have a removable discontinuity at x = 3 since
>
> (1) f is not defined at x = 3.
>
> but
>
> (2) lim (x -> 3) f(x) exists.
>
> The level of discussion is key here.
>
> At the Calculus level, precisely because they want to
> discuss the concept of removable versus non-removable
> discontinuities, the question
>
> "Is f continuous at x = 3?"
>
> is not regarded as a meaningless question.



I was aware of this szenario when I wrote that

>>[...] poles of rational functions are sometimes called
>> "points of discontinuity"[...]


and of course I do not think that the question

(*) "can the continuuous map f: R \ {3} --> R with f(x)=(x^2 - 9)/(x - 3)
be extended to a continuous map g: R --> R ?"

or the question

(**) "does lim_{x->3} (x^2 - 9)/(x - 3) exist? "

are meaningless. My point was that these question should (and can) be
be posed easily within the ordinary definition of a continuity at a
point as used e.g. in the context of analysis and topology.

Meanwhile I also had a look at some calculus books (we do not do
calculus around here, so they are not so common), and indeed you
are correct, a lot of them (but not all) expand the definition of
continuity the way you described:

>[...]
> On the same page, the definition is recast as 3 requirements:
>
> f is continuous at x = a if
>
> (1) f(a) exists
> (2) lim (x -> a) f(x) exists
> (3) lim (x -> a) f(x) = f(a)


Of course no book on analysis would do so, but I did not sample those.
I still think this is unfortunate and it makes me wonder if perhaps
something is wrong with the calculus level.

Let me illustrate this with three variants in the wording of the
definition of continuity at a point:

(V0) f is continuous at a if
(0) f: A --> R is a function
(1) a is in A
(2) lim_(x->a) f(x) = f(a)

(V1) f: A --> R is continuous at a if
(1) a is in A
(2) lim_(x->a) f(x) = f(a)

(V2) f: A --> R is continuous at a in A if
(2) lim_(x->a) f(x) = f(a)

Somebody who used (V0), could happily produce questions like
"is 42 continuous at 3 ?" and answer no (unless 42 is meant to
stand for the function with constant value 42). But most people
would regard such a question as meaningless and prefer to restrict
the scope of the definition at least up to the level of (V1), so that
they could discard "is 42 continuous at 3 ?" as a type error.

My attitude towards (V1) is similar. I find it much clearer to use (V2).
Questions as "is f(x) = (x^2 - 9)/(x - 3) continuous at 3 ?" should
then be regarded as a type error. I thinks that (V2) is less likely
to confuse students and is more in line with the definitions in analysis
and topology. And, as I indicated with (*) and (**) above, it is
sufficient for calculus.

P.S.: in the library I also saw a 'classic edition' (reissued in 1982)
of the 1953 edition of Thomas: Calculus and Analytic Geometry.
I must say that it looks much better than the monster mutant calculus
book that it had grown into in the 1990ies and which was best used as
a paperweight or a door-stopper.

P.P.S.: D.J. Bernstein: Calculus for mathematicians
[ http://cr.yp.to/papers/calculus.(dvi|ps|pdf) ]
is a very good choice as a calculus text. It has only 12 pages.

--
Marc


Date Subject Author
9/28/13
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9/28/13
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