too long since the last math chat

given f(x)=(-1)^x
evaluate f'(x)

unrelated, but posted for similer reasons:

in this thread:
http://www.redhotpawn.com/board/showthread.php?threadid=15717
we determined that 0\0 can take on any one real value for any one situation.

but can it take on an unreal value?

f(x)=x/0

evaluate.

Originally posted by fearlessleader
unrelated, but posted for similer reasons:

in this thread:
http://www.redhotpawn.com/board/showthread.php?threadid=15717
we determined that 0\0 can take on any one real value for any one situation.

but can it take on an unreal value?

f(x)=x/0

evaluate.

What is your definition of the operation denoted by / ?

If it represents division, what is your definition of division?
A typical definition of division is this:

For real numbers w, x, y, and z,

w/x = y/z if and only if z*w = x*y


Now, you define your function as f(x) = f(x)/1 = x/0. From this and the definition of division, it follows that you also have 0*f(x) = 1*x. By the rules of multiplication, it follows that you also have 0 = x.

That is, your function is only defined at one point, namely 0. For any other point, f cannot be defined, for if it were, its own definition would be a contradiction.

But what use is a function whose domain is a single point?

Originally posted by fearlessleader
given f(x)=(-1)^x
evaluate f'(x)

f'(x) = log(-1)*(-1)^x, I suppose.

Originally posted by fearlessleader
given f(x)=(-1)^x
evaluate f'(x)

Is it an analytic complex valued function? If so its derivative is defined everywhere in the complex x-plane. And therefore it should be quite well defined and continuous for all real values of x too.