After reading though these posts, I think if we simply go back to the limit definition of the derivative, we'll see why d(x2)/dx can't be defined as I've defined it.

f(x) = Sum[i=1..x, x]

lim h-->0 (f(x + h)-f(x))/h will get to a closer and closer approximation for smaller and smaller values of h which get ever closer to 0. However, the only values for which h is defined are when h is an integer; the basic value that any limit describes is not fulfilled because the function is not continuous for a finite distance around the point, so the limit itself can't exist in the current form.

However

I got to thinking about this, and thought I'd define f(x) for rational numbers as well.

1.22 = ?
(1+2/10)2=

(A+B/10)2=A2+AB/5+B2/100 = A+A+A...A times + B+B+B...B times/100 + A+A+A...B times / 5

Yay, it works!
1+2/5+4/100 = 1.44 = 1.22

So, let's redefine the function to be
f(x) = f(A, B) = Sum[i=1..A, A]+Sum[i=1..B, B]/100+Sum[i=1..A, B]/5


Granted, this doesn't change the invalidity of differentiating f(x), but it makes it more difficult to spot the mistake

d/dx (f(x)) = d/dA (f(A, B)) + d/dB (f(A, B)) 'Note: partial derivatives there
= 2A + B/5 + 2B + A/5 with the A2 version, and
= (1+1+1... A times) + (1+1+1...B times)/5 + (1+1+1...B times)/100 = A+B/5+B/100 {or A+A/5+B/100 if you feel like it}. These two equations should be equal, and the function is defined at all rational numbers. They of course are not equal to each other, and math is wrong yet again (more or less)!

Note: I may have taken d/dx in terms of A and B wrong, but in any case, they should be equal and they won't be.


Now all that's left is to define f for irrationals, yay!