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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

 Messages: [ Previous | Next ]
 Hetware Posts: 148 Registered: 4/13/13
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Oct 20, 2013 1:33 PM

On 10/20/2013 3:54 AM, quasi wrote:
> Hetware wrote:
>> quasi wrote:
>>>
>>> On the other hand, suppose your proposed question is reworded
>>> in the following way ...
>>>
>>> Given:
>>>
>>> f(x) is defined and continuous for all x in R.

>>
>> I believe we can drop the "defined" since it is implied
>> by "continuous".

>
> Yes.
>

>>> Then if g is defined by
>>>
>>> g(x) = limit (s -> x) f(s)
>>>
>>> can g(x) be used to determine the value of f(x) for all x in R?
>>>
>>> The answer is yes -- in fact, f(x) = g(x) for all x in R.

>>
>> What is your opinion of the definition given as follows:
>>
>> Let f(x) be continuous over R, and f(x) = x/x for all x in R
>> where x/x is a determinate form?

>
> I wouldn't use the phrase:
>
> "where x/x is a determinate form"
>
>
> "where x/x is defined"
>
> You could also say it more simply this way:
>
> "Let f be a function which is continuous over R and such
> that f(x) = x/x for x != 0."
>
> Or even simpler:
>
> "Let f(x) = 1.
>
> But at this point, you surely understand the terminology and
> conventions relating to this issue as used in the text by
> Thomas. Why not just go on from there?
>
> quasi
>

I finally resolved my confusion. I was treating the proposition that f
is continuous to be equivalent to defining f to be continuous.

The reason I spent so much time with this is because my mind
automatically removed the removable discontinuity when I saw the
definition f(t) = (t^2-9)/(t-3). I needed to understand why I could see
f(t) as continuous. I now understand that I was implicitly imposing

I had come to that conclusion about a week ago, then I tried to
formalize my understanding, and got confused with the distinction
between a proposition and a definition.

It didn't help that others were insisting that I could not assert
continuity as part of a definition.

This was posted elsewhere in this thread earlier today. It's how I
finally resolved my confusion.

Let P(f) <=> f(x) is continuous for all x in R.
Let D(f) <=> f(x) has a finite value for all x in R.
Let L(f) <=> limit[f(s), s->x] exists for all x in R.
Let Q(f) <=> f(x) = limit[f(s), s->x]

P <=> D & L & Q
Q => D

Let f(x) be in R for all x in R other than x=0, f(0) = 0/0 and
limit[f(s), s->x] exist for all x in R.

D(f) => F
L(f) => T
Q(f) => F
!P(f) <= F & T & F

Let g(x) be a continuous function on R such that g(x) = f(x) when f(x)
is a real number.

D(g) => T
L(g) => T
Q(g) => T
P(g) <= T & T & T

So quasi was correct. I can define a function g(x) to be continuous,
but it is not the same function as the function f(x) used to define g(x)
everywhere f(x) is defined.

Date Subject Author
9/28/13 Hetware
9/28/13 Michael F. Stemper
9/28/13 scattered
9/28/13 Hetware
9/28/13 quasi
9/28/13 Hetware
9/28/13 quasi
9/28/13 Peter Percival
9/29/13 quasi
9/28/13 Hetware
9/28/13 Richard Tobin
9/28/13 Hetware
9/28/13 tommyrjensen@gmail.com
9/29/13 Hetware
10/6/13 Hetware
10/6/13 Peter Percival
10/6/13 Hetware
10/6/13 quasi
10/8/13 quasi
10/7/13 Peter Percival
9/29/13 Michael F. Stemper
9/29/13 Hetware
9/29/13 quasi
9/29/13 Hetware
9/29/13 magidin@math.berkeley.edu
10/6/13 Hetware
10/6/13 magidin@math.berkeley.edu
10/7/13 Hetware
10/7/13 LudovicoVan
10/7/13 Peter Percival
10/8/13 magidin@math.berkeley.edu
10/12/13 Hetware
10/12/13 fom
10/13/13 magidin@math.berkeley.edu
10/13/13 Richard Tobin
10/13/13 Hetware
10/13/13 Peter Percival
10/13/13 fom
10/13/13 magidin@math.berkeley.edu
10/13/13 magidin@math.berkeley.edu
10/8/13 quasi
10/8/13 magidin@math.berkeley.edu
10/8/13 quasi
10/8/13 quasi
10/12/13 Hetware
10/13/13 quasi
10/13/13 Peter Percival
10/9/13 magidin@math.berkeley.edu
10/9/13 fom
10/10/13 magidin@math.berkeley.edu
10/10/13 fom
10/7/13 Peter Percival
10/7/13 Hetware
10/7/13 fom
10/7/13 Peter Percival
9/29/13 quasi
9/30/13 Peter Percival
9/30/13 Peter Percival
9/30/13 Peter Percival
9/30/13 RGVickson@shaw.ca
9/30/13 Roland Franzius
9/30/13 Richard Tobin
9/30/13 RGVickson@shaw.ca
9/28/13 Peter Percival
9/28/13 Hetware
9/29/13 Peter Percival
9/28/13 Virgil
9/29/13 quasi
9/29/13 Virgil
9/29/13 Hetware
9/29/13 quasi
9/29/13 Hetware
9/29/13 LudovicoVan
9/29/13 quasi
9/29/13 Virgil
9/29/13 magidin@math.berkeley.edu
9/29/13 Peter Percival
9/29/13 FredJeffries@gmail.com
9/30/13 Hetware
9/30/13 magidin@math.berkeley.edu
10/6/13 Hetware
10/6/13 Peter Percival
10/6/13 Peter Percival
10/6/13 magidin@math.berkeley.edu
10/6/13 Peter Percival
10/6/13 magidin@math.berkeley.edu
10/6/13 David Bernier
9/29/13 Peter Percival
9/28/13 Hetware
9/29/13 Richard Tobin
9/30/13 Ciekaw
9/30/13 Robin Chapman
9/30/13 Virgil
9/30/13 LudovicoVan
9/30/13 LudovicoVan
10/6/13 Hetware
10/7/13 Robin Chapman
10/7/13 David Bernier
10/7/13 Hetware
10/7/13 LudovicoVan
10/8/13 Hetware
10/9/13 Peter Percival
10/9/13 Richard Tobin
10/7/13 Peter Percival
10/8/13 Hetware
10/8/13 Virgil
10/8/13 Hetware
10/9/13 magidin@math.berkeley.edu
10/9/13 Peter Percival
10/10/13 Ciekaw
10/9/13 Peter Percival
10/10/13 Tim Golden BandTech.com
10/13/13 Hetware
10/13/13 Peter Percival
10/13/13 Hetware
10/14/13 Peter Percival
10/13/13 Hetware
10/13/13 fom
10/13/13 Hetware
10/13/13 fom
10/14/13 fom
10/14/13 Hetware
10/14/13 magidin@math.berkeley.edu
10/14/13 magidin@math.berkeley.edu
10/14/13 Peter Percival
10/14/13 Hetware
10/14/13 quasi
10/16/13 @less@ndro
10/16/13 quasi
10/19/13 Hetware
10/19/13 quasi
10/19/13 Hetware
10/20/13 fom
10/20/13 quasi
10/20/13 Hetware
10/20/13 fom
10/20/13 Hetware
10/20/13 Peter Percival
10/20/13 Richard Tobin
10/20/13 Hetware
10/30/13 @less@ndro
10/19/13 Hetware
10/10/13 Ronald Benedik
10/10/13 Peter Percival
10/10/13 Virgil
10/18/13 Hetware
10/19/13 Peter Percival
10/19/13 fom
10/19/13 Peter Percival
10/19/13 Hetware
10/19/13 Peter Percival
10/19/13 Hetware
10/19/13 fom
10/19/13 magidin@math.berkeley.edu
10/19/13 Hetware
10/19/13 magidin@math.berkeley.edu
10/20/13 Hetware
10/20/13 quasi
10/20/13 quasi
10/20/13 Hetware
10/20/13 Peter Percival
10/20/13 magidin@math.berkeley.edu
10/20/13 Hetware
10/20/13 Arturo Magidin
10/20/13 Hetware
10/20/13 magidin@math.berkeley.edu
10/19/13 fom