Please rid me of this goto

[Timon Gehr via Digitalmars-d]

On 24.06.2016 02:14, H. S. Teoh via Digitalmars-d wrote:
> 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),

Which just means that there is no limiting value for that point.

> 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.
> ...

No disagreement here. Nothing about this is 'arbitrary' though. All 
notation is convention, but not all aspects of notations are arbitrary.

>
> [...]
>>> 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?
> ...

There exists a function that agrees with + on all values except (2,2), 
where it is 5. If we call that function '+', we can still do algebra on 
real numbers by special casing the point (2,2) in most theorems, but we 
don't want to.

> 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.

Yup.

> At best it is undefined, since it's an essential discontinuity,

Nope. x=0 is an essential discontinuity of sgn(x) too, yet sgn(0)=0.

> just like x=0 is an essential discontinuity of 1/x.

That is not why 1/0 is left undefined on the real numbers. It's a 
convention too, and it is not arbitrary.

> 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.
>...

This is the 21st century and it has become clear what 0^0 should be. 
There is no value in discrediting the convention by calling it 
'arbitrary' when it is not.