Marko, your argument is "this function x**y(a, x) must be continuous on [0, inf), and to be continuous at 0, 0**0 must be a". Since there are many possible values of a, this is not a "justification", this is a proof by contradiction that the premise was faulty: x**y(a, x) doesn't have to be continuous after all.
0**0 is 1, which makes some functions continuous and some functions not, and who cares? It's 1 because that's what is demanded by combinatorial definitions of exponentiation, and its origins in the domain of the natural numbers. Knuth says that thought of combinatorially on the naturals, x**y counts the number of mappings from a set of x values to a set of y values. Clearly there's only one mapping from the empty set to itself: the empty mapping. Number theory demands that performing multiplication among an empty bag of numbers gives you the result of 1 -- even if the empty bag is an empty bag of zeroes instead of an empty bag of fives. The result does not change. Either of those ideas about exponentiation can be thought of as descriptions of its behavior, or as definitions. They completely describe its behavior on the naturals, from which we derive its behavior on the reals. -- Devin On Thu, Jan 8, 2015 at 11:28 PM, Marko Rauhamaa <ma...@pacujo.net> wrote: > Devin Jeanpierre <jeanpierr...@gmail.com>: > >> If 0**0 is defined, it must be 1. > > You can "justify" any value a within [0, 1]. For example, choose > > y(a, x) = log(a, x) > > Then, > > lim y(a, x) = 0 > x -> 0+ > > and: > > lim[x -> 0+] x**y(a, x) = a > > For example, > > >>> a = 0.5 > >>> x = 1e-100 > >>> y = math.log(a, x) > >>> y > 0.0030102999566398118 > >>> x**y > 0.5 > > > Marko > -- > https://mail.python.org/mailman/listinfo/python-list -- https://mail.python.org/mailman/listinfo/python-list