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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.

Niels Korpel
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  • The question is why, even though a function being differentiable on $\Bbb C$ implies it's analytic on $\Bbb C$, its being differentiable on $\Bbb R$ doesn't imply it's analytic on $\Bbb R$. Note the second if-then idea weakens both the antecedent and the consequent, and therefore there's no telling whether the second one is also true. – J.G. Oct 15 '19 at 10:04
  • @OscarLanzi "For example, $f(z)=z^2$ is differentiable at 0 but not throughout any region containing 0, so not analytic at 0." Ahem. – Jean-Claude Arbaut Oct 15 '19 at 10:37
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    I think the confusion stems from using the same name (differentiability) to denote very different properties and it is better understood if we think of what differentiability means on $\mathbb{R^2}$ without the complex structure where it is pretty much the same as on the real line with all possible pathologies there – Conrad Oct 15 '19 at 10:51
  • Strictly speaking, a complex function is analytic at a point iff it is differentiable throughout some region that includes this point. For example, $f(z)=|z^2|$ is differentiable at 0 but not throughout any region containing 0, so not analytic at 0. – Oscar Lanzi Oct 15 '19 at 11:08
  • @Jean typo'ed. Missed the absolute value signs. One day we will solve quantum gravity, then we can tackle really hard problems like making SE comments editable by their writers without time limits. – Oscar Lanzi Oct 15 '19 at 11:09

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