Does there exist a non-trivial function $f:D \rightarrow\mathbb{C}$ holomorphic in the the domain $D=\mathbb{C}- A$ where $A=[-1,1]$ such that $\lim_{z \rightarrow w}{f(z)}=0 \ \forall w \in A $?
I tried to use the identity theorem for holomorphic functions to prove that $g=0$ in D where $g$ is the same as $f$ in D and zero otherwise but I wasn't able to find an accumulation point of the zeros of g in D and so I think that theorem may not be applicable in this situation.
No non-trivial holomorphic map exists. Define $$\widetilde f(z)=\begin{cases} f(z) &\text{ if }z\in \Bbb C\backslash[-1,1],\\ 0 &\text{ if }z\in [-1,1].\end{cases}$$Now, $\widetilde f$ is continuous from hypothesis. Then, one can prove that for each triangle $\Gamma$ in $\Bbb C$ we have $\int_\Gamma \widetilde f=0$, so by Morera's theorem $\widetilde f$ is holomorphic on $\Bbb C$. To prove this, note that this can be done if $\Gamma \subseteq \Bbb C\backslash [-1,1]$, as $\widetilde f$ is holomorphic in $\Bbb C\backslash[-1,1]$. But if $\Gamma$ intersects with $[-1,1]$, then we can consider a sequence of triangles $\Gamma_n$ each contained in $\Bbb C\backslash[-1,1]$ and converging to $\Gamma$. So, we have the uniform convergence $\widetilde f\big|_{\Gamma_n}\to \widetilde f\big|_\Gamma$. Hence, $\int_\Gamma \widetilde f=\lim\int_{\Gamma_n}\widetilde f=\lim0=0$, so we are done in this case also.
Now, by identity theorem $\widetilde f\equiv 0$ on $\Bbb C$ .