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Why can't a function $f : \mathbb{C} \rightarrow \mathbb{C}$ be holomorphic if it isn't continous at at least one point?

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One way to answer this question is using Eric Towers' answer: holomorphic functions are differentiable, and differentiable functions are continuous, so if a function is not continuous, it is not holomorphic.

Another way is to prove it by definition. Take a point $a$ such that $f$ is not continuous in $a$. Now take a sequence $z_n$ such that $\lim z_n=a$ and $|f(z_n)-f(a)|>\delta$ for some $\delta$ (such a sequence exists, because $f$ is not continuous at $a$. Then, the limit $$\lim_{z\to a}\frac{f(z)-f(a)}{z-a}$$ does not exist because $$\frac{|f(z)-f(a)|}{|z-a|} > \frac{\delta}{|z-a|}\to\infty $$ as $z\to a$.

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Holomorphic implies differentiable (at every point of its domain) implies continuous (at every point of its domain).

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