Analyticity means the function locally behaves like (infinite-degree) polynomial. I’d say, it’s not strange that infinitely differentiable functions are not necessarily analytic - polynomials are quite restrictive, after all.

Well, over complex plane, differentiability = infinite differentiability = analyticity, so there’s that.

I wonder if I kinda get it because I have some basic background in electronics, or maybe I don’t get it at all but I think I do? I haven’t done actual math stuff in years