Does continuity of $f(z)$ imply that of $\overline{f(\overline{z})}$?

Question

I was stuck on a problem with complex-valued functions. Here is the question:

If a function $f(z)$ is continuous at a point $z=z_0$, then does this imply that the induced function $\overline{f(\overline{z})}$ is continuous at $z=z_0$? Much help required. Thank you so much!!

Answer 1

The answer is no, in fact the induced function isn't even guaranteed to be defined at that point. For example take a look at the function $f(z)=\frac{1}{z-i}$. $f(z)$ is continuous at $-i$, but $\overline{f(\overline{z})}=\frac{1}{z+i}$ is not continuous at $-i$.