# Why does 'not continous' will lead to 'not holomorphic' for a function

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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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 $z\to a$.