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**ThePerfectHacker** My Complex Analysis book states the following interesting theorem:

Let $\displaystyle N(c,r)$ be a open disk in $\displaystyle \mathbb{C}$. Let $\displaystyle f$ be a complex function whose domain contains the open disk. Let $\displaystyle f(x+iy) = u(x,y)+iv(x,y) \mbox{ for }x+iy \in N(c,r)$. If the parital derivaties $\displaystyle u_x,u_y,v_x,v_y$ all exist and are continous on $\displaystyle N(c,r)$ and at $\displaystyle c \in N(c,r)$ the Cauchy-Riemann equations are satisfied then $\displaystyle f$ is differenciable at $\displaystyle c$.

This is a partial converse for the necessay Cauchy-Riemann condition.

Here is how the proof begins.

Let $\displaystyle c=a+bi$ and let $\displaystyle h,k$ be so that $\displaystyle |h+ik|<r$ (that is they lie in the open disk).

Then,

$\displaystyle u(a+b,b+k)-u(a,b) = [u(a+h,b+k)-u(a,b+k)] + [u(a,b+k)-u(a,b)]$

By the Mean Value Theorem,

$\displaystyle =hu_x(a+x_0h,b+k) + kv_y(a,b+y_0k)$ where $\displaystyle x_0,y_0\in (0,1)$.

And it goes on....

My problem is that, they did not use the MVT properly. Do you agree? (The problem is that this is one of those books which likes to skip steps).