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An example, from general relativity: "If you're on a spaceship, and carry a really good clock down near to the event horizon of a black hole, and come back, you'll find that time has slowed down relative to people who didn't go on the trip". Dwell on that, & it's mindblowing.
Or an example from complex analysis: suppose you have a function that's analytic everywhere in the complex plane. Than integrate it round a closed curve in the complex plane, and you'll find that you get 0. Again, it's mindblowing!
For the last, you need to know what analytic means: (f(z+epsilon)-f(z))/epsilon approaches a constant limit as epsilon approaches 0, everywhere. Intuitively, the key thing going beyond real differentiability is that epsilon can approach 0 from any direction in the complex plane
Mindblowing or not really depends on one’s preconception in the field. In this case integral of a vector field around a closed loop. On a related note, I find it mindblowing that an analytic function is fully determined (or encoded) by its values inside any tiny neighborhood.
I was about to make a similar comment to the first part of this one. Another result one might choose is that if a function is complex-differentiable once, then it is complex-differentiable twice (and hence infinitely many times).
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