In Bak and Newman's Complex Analysis book, the following proposition is stated and proven: Suppose fx and fy exist in a neighborhood of z. Then if fx and fy are continuous at z and fy = i fx there, f is differentiable at z. (statement and beginning of proof)
This proof separates $f$ into $u+iv$ and applies the Mean Value Theorem to the two difference quotients $\frac{u(z+h) - u(z)}{h}$ and $\frac{v(z+h) - v(z)}{h}$. Why can't this logic be directly applied to the difference quotient involving $f$ considered as a function of two real variables?