My Complex Analysis book states the following interesting theorem:

Let

be a open disk in

. Let

be a complex function whose domain contains the open disk. Let

. If the parital derivaties

all exist and are continous on

and at

the Cauchy-Riemann equations are satisfied then

is differenciable at

.

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

Here is how the proof begins.

Let

and let

be so that

(that is they lie in the open disk).

Then,

By the Mean Value Theorem,

where

.

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).