We know two facts:
-A complex function is complex analytic if and only if it is once complex- differentiable.
-A real function is not necessarilty real analytic if it is once real-differentiable, and not even when it is $\infty$ce real-differentiable.
This both makes sense and doesn't.
It makes sense, because of the concept of structure. Complex differentiability is more difficult to comply with, since all possible directions of evaluating the limit need to agree. Therefore once complex-differentiable or holomorphic functions have more structure (we know more about them), and as such these functions are more constrained and comply with stronger properties than real functions do.
But it also doesn't make sense, because of the correspondence principle. The correspondence principle states that more comprehensive theories simplify to a simpler theory when restricted to the circumstances in which the simpler theory was shown to work. Because $\mathbb{R}\subset \mathbb{C}$, the principle seems to apply: we'd expect that the behaviour of complex numbers becomes the behaviour of real numbers when we restrict ourselves to the reals. And so we expect complex functions to behave like real functions when we demand that the only input and outputs are real numbers. And yet, here we have structure in $\mathbb{C}$ that does not imply the same structure in $\mathbb{R}$.
Does anyone know how this paradox is resolved.