OK, you think that my 0^0 = 1 is totally arbitrary. Yes, it is true that 0^x is discontinuous at 0, but that does not mean it is undefined. As I said there are a number of reasons and they would be easier to state if I could remember the various codes, so bear with me. First off, and the most important to me is that when m and n are natural numbers, n^m is the number of functions from m to n. Here I take the usual definition of n as the set {0,1,…,n-1} and then 0 is the empty set. The number of functions from the empty set to the empty set is exactly 1. Second (although not unrealted) is the fact that if A and B are disjoint finite sets and if {x_a|a in A} and {x_b|b in B} are A and B indexed families and if C is the union of A and B, then the product of all the x_c, where c ranges over the union of A and B is simply the product of the x_a times the product of the x_b. When A (or B) is empty, this implies that the empty product has to be defined to be 1. 0^0 is an empty product (of no copies of 0). This, BTW, is another explanation for why 0! = 1. A similar argument implies that the empty sum is 0. Third the usual inductive definitions of addition, multiplication, and exponentiation of integers are (S(n) is the successor of n, the next integer after n; it would be n + 1 if + were already defined):

addition: m + 0 = m and m +S(n) = S(m+n)

multiplication: m*0 = 0 and m*S(n) = m*n + n

exponentiation: m^0 = 1 and m^{S(n)} = (m^n)*m

If 0^0 were anything other than 1, the third one would require a more complicated definition. The situation might be different if exponentiation were defined by induction on the base rather than the exponent, but such a definition would be hopelessly complicated.

So there are three reasons why this is the best definition. I could probably find more if pressed. To say that 0^0 is undefined is to say that no one has defined it. Well, I and many many other mathematicians have defined it and we defined it to be 1. You want to define it otherwise, go right ahead, but unless you can find a good reason, others will not go along. Now suppose you do find a good reason. Not impossible, maybe not even unlikely, although nothing comes to mind. Then you right a paper and you say at the outset that for the purposes of this paper, 0^0 will be understood to be defined as 7. No problem. People are forever starting papers in this way and, so long as there is some reason for doing so, there is no harm.

Someone above said that 0^0 is UNDEFINED. As those there were, somewhere, a compendiium of definitions (and undefinitions) that everyone must follow. Well, there ain’t such. Not even a list of standard notations. Maybe pi is used in one place as the ratio of the circumference to the diameter of a circle. The next place might use for a projection operator (say in some Hilbert space). In a third, it might be an element of a set called P. Maybe I will have e = mc^2 is some paper in which e, m, and c have meanings unrelated to their meanings in physics.