The point is that a function that is differentiable at every point on anopenset (the "open disc" that you refer to) is analytic on that set. And, no "coordinate" does not mean "real". You used the phrase "coordinates axes" which is plural! Both real and imaginary axes are "coordinate axes".

OR, is it that, there is some point where the function is differentiable, but it is not an entire function?

i found the C-R Equations.

du/dx = 3x^2 + 3y^3 -3 , dv/dy = 3y^2 - 2 + 3x^2

du/dy = 6xy , dv/dx = 6xy

du/dx = dv/dy

and du/dy = - dv/dx where x = 0 or y = 0

and thus C-R equations are not satisfied on an open disk, but only on the line x=0 or y=0

I think this is wrong.

Any help appreciated, especially if you point out the thought process required here.

Thanks.