Prove that if a complex-valued function is real differentiable, then limit of line integral around circle exists I've been working on the problem below and really haven't gotten anywhere. For part (1) I've tried the function $f(z) = \bar{z}^2 / z$ but couldn't get it to work and for part (2) I've not sure how to proceed. Any help would be appreciated. Thanks. 
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 each $z \in D$ let
$$ A(z) = \lim\limits_{r \to 0} \frac{1}{2\pi i r^2} \int_{C(z,r)} f(\zeta) d\zeta $$
if the limit exists.
(1) Give an example of a continuous function $f$ defined on a domain $D$ such that $A(z)$ does not exist for some $z \in D$.
(2) Show that if $f$ is $\mathbb{R}$-differentiable on $D$, then $A(z)$ exists for $z\in D$. Find the limit.
 A: Let us put $z=0$ for simplicity. Surely, we can translate our results to work for other $z$.
Write $$f(\zeta)=f(0)+\frac{\partial f}{\partial z}\zeta+\frac{\partial f}{\partial\bar z}\bar\zeta+o(|\zeta|^2),$$
with the partial derivatives evaluated at $0$. Integrate around a circle of radius $r$ centered at the origin and let $r\to0$. Note that $\zeta=re^{it}$ gives $d\zeta=ire^{it}$, and you quickly find that the integral evaluates to
$$\frac{\partial f}{\partial\bar z}+o(1).$$
This answers (2), and probably gives a hint for the search for a solution to (1):
I would try
$$f(z)=\frac{\bar z}{\sqrt{|z|}},$$
for example.
I should have added this explanation of the partials:
$$ df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy
 =\frac12\Bigl(\frac{\partial f}{\partial x}d(z+\bar z)-i\frac{\partial f}{\partial y}d(z-\bar z)\Bigr).$$
Collect terms and insist on having $$df=\frac{\partial f}{\partial z}dz+\frac{\partial f}{\partial \bar z}d\bar z,$$ and conclude
$$\frac{\partial f}{\partial z}=\frac12\Bigl(\frac{\partial f}{\partial x}-i\frac{\partial f}{\partial y}\Bigr),\quad 
\frac{\partial f}{\partial \bar z}=\frac12\Bigl(\frac{\partial f}{\partial x}+i\frac{\partial f}{\partial y}\Bigr).$$
