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Call a function $f : \mathbb{R} \rightarrow \mathbb{R}$ nearly analytic iff it is smooth, and for each point $x \in \mathbb{R}$, either $f$ is analytic at $x$, or there exist $a,b > 0$ such that $f$ is analytic on $(a,x)$ and also on $(x,b)$.

Not every smooth function is nearly analytic.

Nevertheless, the standard examples of smooth-but-nonanalytic functions tend to be nearly analytic. In particular, bump functions are nearly-analytic, suggesting that the theory of "nearly-analytic manifolds" should be very similar to the theory of smooth manifolds.

Question. Is there an accepted name for such functions, and has there been any research into them (e.g. into "nearly analytic manifolds" etc.)

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  • $\begingroup$ We obviously have some low hanging fruit: clearly we can associate with each point $x$ two radii of convergence in which we will get convergent power series (namely $\frac{a+x}{2}$ and $\frac{b+x}{2}$) Moreover, every analytic function is nearly analytic. $\endgroup$ Sep 4, 2018 at 6:03
  • $\begingroup$ Other than those and other obvious properties, you mention that the theory of "nearly analytic manifolds" should be very similar to the theory of smooth manifolds. This is of course so, since every nearly analytic function is smooth. I then ask: what kinds of results are you looking for with "nearly-analytic" functions? $\endgroup$ Sep 4, 2018 at 6:05
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    $\begingroup$ @BrevanEllefsen, I don't have anything specific in mind. I'm just a curious observer who likes to know the lay of the land :) $\endgroup$ Sep 4, 2018 at 6:12

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