Hetware wrote: > On 10/6/2013 11:51 PM, Arturo Magidin wrote:
>> repeating it, well, you can guess what that implies... >> > > That you have little of value to contribute. The context was reasonable > clear. If you wanted clarification, you should have asked for it.
Ho ho ho.
>>> In this case, the domain is all real numbers were f(t) is defined. >> >> That was the point. But you are trying to extend this from "where it >> is defined" to "wherever I can define it irrespective of the formula >> given". >> >> I say right after: >> >>>> Because this becomes both onerous and complicated, there is a >>> >>>> standard convention that is, I am positive, mentioned in your >>>> book. >>> >>>> This convention is: >>> >>>> >>> >>>> If a function is described by giving a formula, and no domain is >>> >>>> explicitly specified, then it is agreed that the domain of the >>> >>>> function is the natural domain: that is, the domain is the set of >>>> all >>> >>>> numbers for which the expression, *as given*, makes sense. > > Ah, but there's the rub. It did make sense to me to treat the function > as defined where t=3.
It hardly makes sense to "treat" something as if it had a property that it doesn't.
> Until I was able to sort out the axiomatic > ordering of assumption, I was confused by applying an assumption that is > generally applicable in the domain where I apply mathematics. > >>> and the observation that >>> >>> (t^2-9)/(t-3) is meaningful for all t!=3 dictates that f(3)=6. >> >> Nope. Because >> >> (i) "f(t) is continuous" does not mean the same thing as "f(t) is >> continuous everywhere"; and > > At this point, I already stated that it is continuous for all real > values t. You simply chose to ignore that fact.
f is not continuous at 3 because it isn't even defined there.
>> (ii) Because by asserting that f(t) is continuous you are ASSUMING >> that f is defined at 3. > > You are finally catching on to what I was saying. > > However, whether or not f(t) is defined at 3 >> is *precisely* the crux of the matter here. > > I defined it as continuous.
You don't define a function as continuous. You define it and then deduce that it is continuous if it is so.
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