Hi ppl, would love the solutuin to "show that f(Z) = Z(hat) is not differentiable"
Or you could use the reasoning that leads to the Cauchy-Riemann equations. Let z= x+ iy where x and y are real numbers. Then $\displaystyle \overline{z}= x- iy$. To find the derivate at, say $\displaystyle z_0= x_0+ iy_0$ we form the "difference quotient" $\displaystyle \frac{f(z)- f(z_0)}{z- z_0}=\frac{x- iy- (x_0- iy_0)}{x+ iy- (x_0+ iy_0)}= \frac{x- x_0+ i(-y- y_0)}{x-x)+ i(y- y_0)}$ and take the limit as z goes $\displaystyle z_0= x_0+ iy_0$. Since that is a two dimensional limit, in order that the derivative exist, the limit must be the same as we approach from any direction.
In particular, what do we get if we approach $\displaystyle (x_0, y_0)$ along the line $\displaystyle (x, y_0)$ ?
What do we get if we approach $\displaystyle (x_0, y_0)$ along the line $\displaystyle (x_0, y)$? Are those the same?