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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 13, 2013 3:32 PM

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.

You cannot prove a definition.

>> 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?

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