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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 8:49 PM

On 10/13/2013 4:54 PM, Peter Percival wrote:
> 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?
>

I appreciate your objection. The same thing has been in my mind. It is
a question of the chicken and the egg.

Let f(t) = (t^2-9)/(t-3) where t!=3, and f(3) = limit[(t^2-9)/(t-3),
t->3]. Is, by this definition, f(t) a continuous function over the set
of real numbers t? For the sake of further exposition, I will answer for
you in the affirmative.

I believe we have agreed that (t^2-9)/(t-3) is continuous everywhere
except for t=3. When f(3) is defined as

f(3) = limit[(t^2-9)/(t-3), t->3],

We have arrived at a function which is defined for all real numbers. So
now we have a function f(t) which is defined for all real numbers t, and
is globally continuous.

If we assert that f(t) is continuous, we are stipulating that

f(t) = limit[(f(s), s->t]

for all real numbers t. The function defined as

g(t) = limit[(s^2-9)/(s-3), s->t]

is identical to f(t) as defined above. The difference in connotation is
that the definition g(t) is not a piecewise definition.

continuous and then being able to show that it is not continuous. That
results in a contradiction. You get your hiney spanked, and you are
expelled from Plato's Heaven, and sent to the purgatory of teaching

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