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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 ]
 Peter Percival Posts: 2,623 Registered: 10/25/10
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Oct 13, 2013 4:54 PM

Hetware wrote:
> On 10/9/2013 4:49 AM, Peter Percival wrote:
>> Hetware wrote:
>>> On 10/7/2013 8:39 PM, Peter Percival wrote:
>>>> Hetware wrote:
>>>>> On 10/7/2013 4:56 AM, David Bernier wrote:
>>>>>> On 10/07/2013 03:21 AM, Robin Chapman wrote:
>>>>>>> On 07/10/2013 04:34, Hetware wrote:
>>>>>>>> On 9/30/2013 4:03 AM, Robin Chapman wrote:
>>>>>>>
>>>>>>>>> Hetware: 0/0 = 3
>>>>>>>>>
>>>>>>>>> Ciekaw: 0/0 = 1
>>>>>>>>>
>>>>>>>>> Any more entrants?
>>>>>>>>>

>>>>>>>>
>>>>>>>> To be correct; Hetware: 0/0 = 1 (under certain circumstances).

>>>>>>>

>>>>>>
>>>>>> Heware, if 0/0 = 1, then 0/0 = (100*0)/0 = 100*(0/0) = 100*1 = 100.
>>>>>>
>>>>>> So, assuming 0/0 = 1, we find that 1 = 100 :(
>>>>>>
>>>>>> David
>>>>>>

>>>>>
>>>>> That statement came with a qualification. That is, given a function
>>>>> defined by f(t) = (t^2-9)/(t-3), I could assume (t-3)/(t-3) = 1, even
>>>>> where t=3.

>>>>
>>>> "given a function defined by" is irrelevant. At t=3 (t-3)/(t-3) is
>>>> undefined.
>>>>

>>>>> I've already shown that a modified version of that
>>>>> proposition does make sense.

>>>>
>>>> No you haven't.
>>>>

>>>>> Given a function f(t) continuous for all real numbers t and defined by
>>>>> (t^2-9)/(t-3) everywhere the expression is meaningful, that
>>>>> function is
>>>>> identical to g(t) = (t+3). My original mistake was to assume
>>>>> continuity
>>>>> after using (t^2-9)/(t-3) to define the entire function.

>>>>
>>>> What you wish to say is that the function g:R->R defined by
>>>>
>>>> g(t) = f(t) if t=/=3
>>>> = 6 otherwise
>>>>
>>>> has these properties:
>>>> i) g = f where f is defined,
>>>> ii) g is defined on the whole of R,
>>>> iii) g is continuous.
>>>>
>>>>
>>>>

>>>
>>> The value of 6 at t=3 follows from the stipulation of continuity.

>>
>> The value of what? f or g? f has no value at 3.
>>

>
> Given a function f(t) continuous for all real numbers t and defined by
> (t^2-9)/(t-3) everywhere the expression is meaningful, that function is
> identical to g(t) = (t+3).
>

>>> It is
>>> meaningful to say that f(t) is continuous over the domain of real
>>> numbers.

>>
>> So what? Is it true? Yes, by proof, not by stipulation.

Whoops. I didn't mean to claim that f is continuous at every real.

> You cannot prove a definition.

To prove that f is continuous throughout its domain (which is what I
meant) is not proving a definition.

Is this the situation: "Here is a function f not defined at every real.
Can I find a function g with these properties: (i) it agrees with f
where f is defined, and (ii) it continuous at every real?" The answer
is yes.

>>> It is also meaningful to say that f(t) = (t^2-9)/(t-3)
>>> everywhere that the rhs is meaningful.

>>
>> I would say "f is defined everywhere (in R) that (t^2-9)/(t-3) is
>> defined".

>
> And you would not be giving the same definition as I am giving.
>

>>> That is sufficient information
>>> to determine that f(3) = 6.

>>
>> f(3) doesn't equal anything. One may "fill the gap" by defining g as I
>> have done above, *then* g(3) = 6. But g isn't f.
>>

>
> "Definition 2-1.1. A function f is a set of ordered pairs, no two of
> which have the same first element. The set of first elements of the
> pairs is called the domain of the function, whereas the set of second
> elements of the pairs is called the range. The domain and range
> elements are related by a given rule"
>
> The definition of continuity provides the rule for determining the value
> of the function f(3) which is *DEFINED* as continuous, and is given by
> f(t) = (t^2-9)/(t-3) where t!=3. By definition f(3) has a definite,
> finite value at f(3) and that value is f(3) = limit[(t^2-9)/(t-3), t->3].
>
> Where do you get the idea that one cannot define a function as
> continuous? How would you prove, for example, the mean value theorem,
> if you could not define an arbitrary function to be continuous?

I can define an arbitrary function to be continuous? What if my chosen
function is not continuous?

--
The world will little note, nor long remember what we say here
Lincoln at Gettysburg

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