Suppose $u:\mathbb{D}\to\mathbb{R}$ is a continuous function, where $\mathbb{D}$ is the open unit disk. Let $I$ be a closed segment in $\mathbb{D}$. To make things simpler we can suppose $I=[a,b]$ is a closed interval in $\mathbb{R}\cap\mathbb{D}$. Suppose that $u:\mathbb{D}\setminus I\to\mathbb{R}$ is smooth. Can $u$ extend to being smooth on all of $\mathbb{D}$? If not, what extra conditions could we put on $u$ to make this work?
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