On continuous extension of harmonic function I am asked to give a specific example of bounded harmonic function defined on $\{|z|<2\}\setminus[-1,1]$ which cannot be continuously extend to $\{|z|<2\}$.
It's well know that if a bounded harmonic function is defined on a punctuated disc with only single point deleted then it can be harmonically extended. I am trying to find some analytic function whose real part is bounded over $\{|z|<2\}\setminus[-1,1]$ while the imaginary part is not, but it does not come to me very quickly. Any comment will be greatly appreciated! Thank you!

Hint: Consider the real or imaginary part of $\sqrt{z^2-1}$
This can be well-defined on $\mathbb{C}\setminus[-1,1]$ by computing
$$
\frac12\log(z^2-1)=\frac12\int_{\sqrt2}^z\left(\frac1{w-1}+\frac1{w+1}\right)\,\mathrm{d}w
$$
along any path that doesn't cross $[-1,1]$. The fact that the total residue at $-1$ and $1$ is $1$, means that exponentiating gives a well-defined function. Exponentiating gives $\sqrt{z^2-1}$. Take the real or imaginary part.
