On Fri, Jun 24, 2016 at 01:58:01AM +0200, Timon Gehr via Digitalmars-d wrote: > On 24.06.2016 01:18, H. S. Teoh via Digitalmars-d wrote: > > On Thu, Jun 23, 2016 at 11:14:08PM +0000, deadalnix via Digitalmars-d wrote: > > > On Thursday, 23 June 2016 at 22:53:59 UTC, H. S. Teoh wrote: > > > > This argument only works for discrete sets. If n and m are reals, > > > > you'd need a different argument. > > > > > > > > > > For reals, you can use limits/continuation as argument. > > > > The problem with that is that you get two different answers: > > > > lim x^y = 0 > > x->0 > > > > but: > > > > lim x^y = 1 > > y->0 > > ... > > That makes no sense. You want lim[x->0] x^0 and lim[y->0] 0^y.
Sorry, I was attempting to write exactly that but with ASCII art. No disagreement there. > > So it's not clear what ought to happen when both x and y approach 0. > > > > The problem is that the 2-variable function f(x,y)=x^y has a > > discontinuity at (0,0). So approaching it from some directions give > > 1, approaching it from other directions give 0, and it's not clear > > why one should choose the value given by one direction above > > another. ... > > It is /perfectly/ clear. What makes you so invested in the continuity > of the function 0^y? It's just not important. I'm not. I'm just pointing out that x^y has an *essential* discontinuity at (0,0), and the choice 0^0 = 1 is a matter of convention. A widely-adopted convention, but a convention nonetheless. It does not change the fact that (0,0) is an essential discontinuity of x^y. [...] > > not something that the mathematics itself suggest. > > ... > > What kind of standard is that? 'The mathematics itself' does not > suggest that we do not define 2+2=5 while keeping all other function > values intact either, and it is still obvious to everyone that it > would be a bad idea to give such succinct notation to such an > unimportant function. Nobody said anything about defining 2+2=5. What function are you talking about that would require 2+2=5? It's clear that 0^0=1 is a choice made by convenience, no doubt made to simplify the statement of certain theorems, but the fact remains that (0,0) is a discontinous point of x^y. At best it is undefined, since it's an essential discontinuity, just like x=0 is an essential discontinuity of 1/x. What *ought* to be the value of 0^0 is far from clear; it was a controversy that raged throughout the 19th century and only in recent decades consensus began to build around 0^0=1. T -- Let X be the set not defined by this sentence...
