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.)