Cauchy-Riemann equation

Cauchy-Riemann equation?-
statement:-
suppose f(z)=u+iv is differentiable at z=(x,y),then at (x,y)
the first order partial derivative of u and v must
exist ,u_x=v_y and u_y=?-v?_x must be true at (x,y),further.
f ?(z)=u_x+iv_x ,where the partial derivative are evaluated at (x,y).
these partial differentiable equation
i.e u_x=v_y and u_y=-v_x are called cauchy-Riemann equation
note?-
this theorem tells that for the differentiablilty of f at
z=(x,y) ,?u/?x-?v/?y=0 and ?u/?y+?v/?x=0
and we have to proof that if a function defined through out serve neighbourhood of a point (x,y)
,suppose the partial out the neighbourhood and these partial dervative satisfy
cauchy riemann equation at (x,y).in addition if these partial derivative are
continuous at (x,y) then f ?(z) exist.
note:-
the above result will be true ,if instead of the continuity of all the four partial
derivative we are having the continuity of u_x and u_y [or v_x and v_y ]or
[u_x and v_x ]or [u_y and v_y ]