Example where $F$ is not continuous on the closed unit disk

Question

Let $f$ be a continuous function on the unit circle ${\partial D}=z:|z|=1$. Define $$F(z)=\begin{cases} f(z),&\text{if}|z|=1\\\frac{1}{2\pi i}\oint_{\partial D}\frac{f(w)}{w-z}dw, &\text{if} |z|<1 \end{cases}$$

Find an example where $F(z)$ is not continuous on the closed unit disk $D(0,1)={z:|z|\leq1}$

Hint from professor: let $f(z)=\bar{z}$

I just started complex analysis and have no idea how to do this question so I would really appreciate it if anyone can provide a solution.

Answer 1

Hint: Let $f(z)=\bar z.$ Note that if $z=e^{it},$ then $\bar z = e^{-it}.$ Thus for $|z|<1,$

$$F(z) = \frac{1}{2\pi i}\int_0^{2\pi} \frac{e^{-it}\cdot ie^{it}}{e^{it} - z}\,dt.$$