6:21 PM
Theorem in Conway's book on complex analysis: Let $u$ and $v$ be real-valued functions defined on a region $G$ and suppose that $u$ and $v$ have continuous partial derivatives. Then $f : G \to \Bbb{C}$ defined by $f(z) = u(z) +iv(z)$ is analytic if and only if $u$ and $v$ satisfy the Cauchy-Riemann equations.
For the backwards direction, he writes $u(x+s,y+t) - u(x,y) = [u(x+s,y+t)-u(x,y+t)] + [u(x,y+t)-u(x,y)]$
And applies the mean value theorem "to each of these bracketed expressions" to get $u(x+s,y+t) - u(x,y+t) = u_x(x+s_1,y+t)s$ and $u(x,y+t)-u(x,y) = u_y(x,y+t_1)t$ for $s_1,t_1$ such that $|s_1| < |s|$ and $|t_1| < |t|$.
How exactly is he using the mean value theorem to get, for example, $u(x+s,y+t) - u(x,y+t) = u_x(x+s_1,y+t)s$? It appears that he is treating $s,y,$ and $y$ as constants; but I can't figure out how he is applying it.