Please rid me of this goto

[H. S. Teoh via Digitalmars-d]

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...