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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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Hetware

Posts: 148
Registered: 4/13/13
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Oct 19, 2013 3:12 PM
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On 10/16/2013 5:38 PM, quasi 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.
>
> As a reference, if you have access to the text
>
> Stewart
> Calculus - Early Transcendentals, 6th Ed (2008)
>
> On page 119, continuity at x = a is defined this way:
>
> f is continuous at x = a if lim (x -> a) f(x) = f(a)
>
> 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)
>
> On page 120, the following example is given ...
>
> =======================================================
>
> Question:
>
> Let f(x) = (x^2 - x - 2)/(x - 2).
>
> Where is f discontinuous?
>
> Answer:
>
> f is discontinuous at x = 2 since f(2) not defined.
>
> =======================================================
>
> quasi
>


Of course, when taking a calculus test which assumed the widely accepted
definitions and interpretations, I gave the answer you have provided.

Let me propose a test question:

Given f(x) defined over R, limit[f(s), s->x] exist over R, and g(x) =
limit[f(s), s->x] can g(x) be used to find the value of f(x) for all x in R?



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9/28/13
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Hetware
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