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.
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.
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.