As am I, by training if not by profession. As is lotharbot. You're in a thread full of mathematicians. :)
Which is what I would expect on this site, actually. I'm always timid making technical claims here unless I'm sure I'm correct; it seems to be a place frequented by arbitrarily large fish.
To answer your question, if 0^0 = 1 is _the_ answer in discrete setting, and it's _an_ answer in continuous setting, why don't we just agree that 0^0 = 1 and stop creating confusing situation where sometimes it's defined and sometimes it's not.
I'm not persuaded it is always the answer. I think the fact that it is an indeterminate form in limits is a forceful enough demonstration of that. It all depends on context. If I came across a 0^0 in, say, an engineering context, my first instinct would be to check whether the formula was defined in that case, not to just assume that 1 would work.
I mean, it's like 1/0. If you're working in R, that's simply illegal. If you're working in R*, it's the infinite point. If you're taking a limit, it means "unbounded". If you're working in my favorite field, the hyperreals, it could be any number of flavors of infinity depending on the flavor of zero it was.
It would be foolhardy to try to define the symbol; without a context to supply some sort of sense, it is nonsense. And that is how I feel about 0^0 as well.
Please, treat exponentiation just like every other function out there. I don't get the whole limit argument at all. Given any function f: R x R -> R, if it happens that a_n -> a, b_n -> b, but lim f(a_n, b_n) != f(a, b), people just say that f is not continuous in (a, b), and the case is over. However, if f happens to be exponentiation function, people instead argue that f should not be defined in (a, b), forgetting about the fact that the theorem which lets you take a limit of an argument instead of a limit of a function values works only under assumption that f is continuous in a proper point. Instead of noting that there's no contradiction because the assumptions are not satisfied, people just run away from it, declaring 0^0 as undefined.
From this point of view, the whole notion of "indeterminate form" makes just as little sense as distinguishing some arbitrary class of functions and calling them "elementary". Why are some points of discontinuity of some functions more special than other points of discontinuity of other functions? Why sin is more elementary than gamma? Historical heritage of confusion, I guess.
Consider this related case: if you evaluate a limit and you get 0/0, you recognize that you need to do more work to find the actual limit. It could be 1, -1, 0, infinite, etc. depending on how you reached it (say, sin(x)/x versus sin(x)/x^2). The issue is not the continuity of x/x; the issue is whether setting a convention for 0/0 would give you the right value for a limit. Since it doesn't always, we call it "indeterminate".
Similarly, if you're evaluating a limit and you get 0^0 you need to do more work. You can't just stop and say "oh, that's 1". It depends on what function you used to get there -- x^x will give you a different answer from ( e^(-1/x) )^x. Again, it has nothing to do with the continuity of exponentiation. The issue is whether the convention of 0^0=1 is correct in the specific part of mathematics you're working in.
The same argument can be made if you're working in the hyperreals, or if you're working with field axioms -- the convention 0^0 doesn't work in that context.
Please, by all means, use the convention 0^0=1 when it's appropriate. But understand that it's not always appropriate. Not every mathematician works in the particular subschool that you do; not every mathematician is going to find your convention appropriate.
As am I, by training if not by profession. As is lotharbot. You're in a thread full of mathematicians. :)
Which is what I would expect on this site, actually. I'm always timid making technical claims here unless I'm sure I'm correct; it seems to be a place frequented by arbitrarily large fish.
To answer your question, if 0^0 = 1 is _the_ answer in discrete setting, and it's _an_ answer in continuous setting, why don't we just agree that 0^0 = 1 and stop creating confusing situation where sometimes it's defined and sometimes it's not.
I'm not persuaded it is always the answer. I think the fact that it is an indeterminate form in limits is a forceful enough demonstration of that. It all depends on context. If I came across a 0^0 in, say, an engineering context, my first instinct would be to check whether the formula was defined in that case, not to just assume that 1 would work.
I mean, it's like 1/0. If you're working in R, that's simply illegal. If you're working in R*, it's the infinite point. If you're taking a limit, it means "unbounded". If you're working in my favorite field, the hyperreals, it could be any number of flavors of infinity depending on the flavor of zero it was.
It would be foolhardy to try to define the symbol; without a context to supply some sort of sense, it is nonsense. And that is how I feel about 0^0 as well.