Problem: Does there exist a function $f : \mathbb{C} \to \mathbb{C} $ that is holomorphic at every point on the unit
circle $\mathbb{S} ^1 = \{ z\in \mathbb{C} : \vert z \vert = 1 \} $ and not holomorphic anywhere else in the complex plane? If yes, provide such a function with complete justification. If not, explain why not.
Solution: I encourage you to attempt to solve the problem today. The solution will be provided tomorrow. This will give you the opportunity to test your understanding of the problem and to improve your skills in solving similar problems in the future.