Another question on continuous functions and Cauchy's Integral formula. Let $C(z,r)$ denote the circle centered at $z$ with radius $r$. Let $f$ be a continuous function defined on a domain $D$. For $n=1,2$  and each $z \in D$ let 
$A_n(z)=\lim_{r\to 0} \frac{1}{2\pi ir^n} \int_{C(z,r)} f(\zeta) d\zeta$ if the limit exists.
Find an example of a continuous function $f$ for which $A_2(z)$ does not exist for some $z \in D$
$\textbf{My Attempt:}$
Let $\gamma(t)=z+re^{i\theta}$
I have tried a lot of functions which all evaluate to either $0$ or $2\pi i r$
For example let $f(z)=\bar{z}$
$A_2(z)=\lim_{r \to 0} \frac{1}{2\pi i r^2}\int_{0}^{2\pi} (\bar{z}+re^{-i\theta})rie^{i\theta}=\lim_{r \to 0} \frac{1}{2 \pi i r} \int_{0}^{2\pi} \bar{z}e^{i\theta}+r=2\pi r$ and so $A_2(z)$ exists. 
I have also tried $f(z)=z\bar{z}$,$f(z)=Re(z)$, $f(z)=Im(z)$. 
Also, the integral for any analytic $f$ will be zero and I am not sure what to conclude from $\frac{0}{0}$.

Hint: Consider
$$f(z) := \frac{\overline{z}}{|z|^{\frac{1}{2}}}$$
