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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: 10,226
Registered: 7/15/05
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Oct 16, 2013 4:39 PM
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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


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9/28/13
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