A version of Casorati- Weierstrass Theorem?

1. Let $f\in\mathcal{O}(\mathbb{D},\mathbb{D})$. Let $p\in\partial\mathbb{D}$ be such that $f$ is not continuous at $p$. Is it true that for every $r>0$ we have $\overline{f(\mathbb{D}\cap B(p;r))}=\overline{\mathbb{D}}?$ Here $B(p;r)$ denotes the open ball of radius $r$ at $p.$

2. As it turns out (see the answer given by user mrf) that the answer for above question is no untill we put some extra conditions. So here are the extra conditions. Suppose there exist sequences $\{z_\nu^i\}_{\nu\in\mathbb{Z}},\,i=1,2$ converging to $p$ such that $\{f(z_\nu^1)\}_{\nu\in\mathbb{Z}_+}$ converges in $\mathbb{D}$ and $\{f(z_\nu^2)\}_{\nu\in\mathbb{Z}_+}$ converges in $\partial\mathbb{D}.$ The question remains the same as above.

Remark: Note that the condition that $f$ is discontinuous at $p$ in the first part is redundant because of the hypotheses in the second part.

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The answer to revised question is still no. The same example as in Bounded holomorphic function in the Unit disc having essential singularity at a boundary point. works: $$F(z)=\exp((z-1)^{-1})+1-e^{-1/2}$$ maps $\mathbb D$ onto the disk $B(1-e^{-1/2}; e^{-1/2})$. Moreover, for every $r>0$ the image of $B(1;r)\cap \mathbb D$ under $f$ is $B(1-e^{-1/2}; e^{-1/2})$, which implies the existence of $z^i_\nu$ as in the question.

(By the way, it is advisable to avoid using $i$ as an index in complex analysis.)

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Thanks for reminding to avoid using index i. – Abelvikram Jun 22 '13 at 4:52

Trivially no. Take a holomorphic $f$ with $|f(z)| < 1/2$ for all $z$ that is discontinuous at a boundary point.

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Oh yes, infact, it was trivial, thanks. – Abelvikram Jun 19 '13 at 12:58
Actually I asked the problem in a way that it became trivial. In fact I was trying to generalize. In the particular context in which I was interested there was a sequence converging to this point of discontinuity such that the sequence formed by the image of this sequence by "f" converges to the boundary. – Abelvikram Jun 19 '13 at 14:45
I edited the problem. Can you please tell me if it is permissible to unaccept the answer when you make change in the problem? Thanks. – Abelvikram Jun 19 '13 at 15:20
@Abelvikram: I think it would be better to ask a new question instead. mrf's answer no longer makes much sense as an answer to the new version of the question. – Martin Jun 19 '13 at 16:37
Yeah but I wasn't sure about asking it again, as I think the question basically is same. Do you think it would be better to ask it again? – Abelvikram Jun 19 '13 at 16:40