Is it true that if the complex function $\bar{f}$ is differentiable, then $f$ is differentiable?

Question

If $\bar{f}=u(x,y)-iv(x,y)$ is differentiable, then the first-order partial derivatives of $u$ and $-v$ exist and they are continuous. Also, the Cauchy-Riemann equations hold, i.e., $u_x=-v_y$ and $u_y=v_x$, but from these relationships, how can we know $f$ is differentiable?

Answer 1

It is not true. Take $f(z) = \overline z$.

Answer 2

In my text, by Saff &Snyder Complex Math. For Scientists , Engineers and Scientists on analyticity,pg 51,

“Notice that we will have to ban the congugate function $\bar z$ because if we admit we will open the gate to x[=(z+$\bar z$/2)] and y[=(z-$\bar z/2i)]$

$\bar z $=x-iy (inadmissible)