It's worth noting that, unlike a real function, a complex function that is differentiable in a neighborhood is infinitely differentiable in that neighborhood. An informal intuition behind this: in the reals, for a limit to exist, the left and right limit must agree. In C, the limit from every direction must agree. Thus, a limit existing in C is "stronger" than it existing in R.
Edit: wikipedia pages on holomorphism and analyticity (did I spell this right) are good
Watched a debate between Emily M. Bender and Sébastien Bubeck
an OpenAI researcher
from March. As usual, Dr. Bender fucking rules. Bubeck struck me as an idiot and kind of an ass.