Please help me finding counterexample (continuous but not analytic).

Question

Statement)

Say $f : D \to \mathbb{C}f$ by $f(x,y) = u(x,y) + i v(x,y)$

$f(z) \in C(D)$ and $f$ satisfies Cauchy-Riemann equation $\Rightarrow $ $f(z) \in H(D)$

Here the notation, $C(D)$ and $H(D)$ mean the set of the continuous function on $D$ and set of the analytic functions on $D$

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Firstly I would suggest my conclusion is the above statement is false.

Here are the reason why I thought like that. To claim $f(z) \in H(D)$, We need two condition first, satisfying the C.R. equation and second, there are $u_x, u_y, v_x$ and $v_y$ should be all continuous. But, In the above there are just condition that $f(z)$ is continuous not its derivative. Hence my conclusion is it is false.

So I tried to find the counter example, but failed. Does any one help me to find that?

(If my thought is incorrect, it would be appreciated taking out What I've missed.)

Thanks.

Answer 1

If $D$ is an open set in $\mathbb C$ and if C - R equations are satisfied on $D$ then $f \in H(D)$. So you are looking for counter-example that does not exist.

However, if C - R equations are satisfied at a point the function need not be differentiable at that point. You can find an example Rudin's book.