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A complex function $f(z)$ is holomorphic at point $a$ if it’s complex differentiable at $a$. More formally the limit:
\[\lim_{h \rightarrow 0} \frac{f(a + h) - f(a)}{h} = f'(a)\]for $h \in \mathbb{C}$ exists. Without any qualifications, a holomorphic function is a function that is holomorphic at every point on its domain.
Addition, subtraction and multiplication preserve holomorphism. Division preserves holomorphism as long as the divisor is non-zero [1].
Differetiation preserves holomorphism. In fact holomorphic functions are infinitely differentiable [2].