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**ThePerfectHacker** Let $\displaystyle f: \{z\in \mathbb{C}| \Im (z) > 0 \} \mapsto \mathbb{C}$ be continous on $\displaystyle \{ z\in \mathbb{C}|\Im (z) > 0\}$ and analytic on $\displaystyle \{ z\in \mathbb{C}|\Im (z) > 0 \} \setminus \{ z\in \mathbb{C}: |z| = 1\}$ prove that $\displaystyle f$ is analytic on $\displaystyle \{ z\in \mathbb{C} | \Im(z) > 0\}$.

My idea was to transform this problem into a simpler one. If $\displaystyle f$ is analytic on a region minus a line segment and continous there, then it must be analytic everywhere on the region. Thus, perhaps a conformal map will transform the circle into a line segment.