On 10/20/2013 3:32 PM, fom wrote: > On 10/20/2013 12:33 PM, Hetware wrote: >> 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. > > > Good. Glad to hear you think you have it > straight. > > My original response was going to be the remarks > below. Then I was just going to cancel. Then > I changed my mind again. > > You might consider them and see if they make > sense with how you are viewing the situation > now. > > Or not.
My original confusion arose from the observation that f(t) = (t^2-9)/(t-3) bears the seeds of its own continuity, so to speak. That is to say, it provides all the information I need in order to "patch" it and make it continuous. So in some sense, it communicates a definition of a continuous function.
Then I began to appreciate that f(t) does not explicitly define a continuous function. I tried to formalize my thinking by using a method similar to that used to prove sqrt(2) is irrational. That is, assume that it is rational, and then demonstrate a contradiction.
So I assumed f(t) is continuous. To my mind, that meant remove the discontinuity. Once I removed the discontinuity, the function became continuous. But I wasn't testing the proposition that f(x) is continuous, I was redefining f(x).
> And, if any of my posts (or occasional frustrations) > contributed to your confusion. Accept my apologies. > > ==================== > > I tried to explain what had been going on > when I mentioned the distinction concerning > the 'intensional' and 'extensional' distinction. > > Wikipedia has these links: > > http://en.wikipedia.org/wiki/Enumerative_definition > > http://en.wikipedia.org/wiki/Extensional_definition > > http://en.wikipedia.org/wiki/Intensional_definition > > If you read about enumerative definition first, it will > give you the sense behind the definition of function > as a listing of ordered pairs (an enumeration of pairs). > > The elements of the pairs *must* *exist* in the enumeration > and if some term that *might* be used does not refer to any > object in the domain, then that pair *cannot* *exist* in > the enumeration. > > But, one usually speaks of extensional definitions and > the ordered pair specification is considered extensional > since the outright need of an enumeration is irrelevant > to its purpose. > > All "rule-based" definitions of functions are intensional. > > Two "rule-based" definitions may or may not be "the same". > > The role of extensional definition with respect to the > rule-based specification of functions is that it provides > a criterion for when two rule-based definitions refer to > the same extensional specification. If they do not, then > they are not the same. That is, they are not two different > rules asserting the same thing. They are two different > rules asserting two different things. > >