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.
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.
Mathematicians arbitrarily chose its value to be 1 based on arguments
like the one Timon gave, but it's an arbitrary choice,
It is absolutely /not/ 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.
