Some functions of a real variable are infinitely smooth (have derivatives of all orders) but are not analytic (at some points $a$, the Taylor series at $a$ does not represent the function at any interval around $a$). All sources, such as Wikipedia article Non-analytic smooth function, give about the same example: $f(x)=\exp(-1/x)$ or $f(x)=\exp(-1/x^2)$; the first one is extended by $0$ to the interval $(-\infty,0]$, the other only needs to be defined as $0$ at $x=0$.

But I have not found any references to the origin of these standard examples. Who gave the first example of a smooth non-analytic function of a real variable? Was it a version of the above functions, or something different?


1 Answer 1


An excellent source for the early history of smooth non-analytic functions is the paper

Gerald G. Bilodeau. The origin and early development of non-analytic infinitely differentiable functions, Arch. Hist. Exact Sci., 27 (1982), 115-135. MR84g:26017.

Also, Dave Renfro wrote an excellent historical Essay on nowhere analytic $C^\infty$ functions in two parts (with numerous references). See 1 (dated May 9, 2002 6:18 PM), and 2 (dated May 19, 2002 8:29 PM).

The first example is Cauchy's (in 1823), and is indeed $e^{-1/x^2}$. The point $0$ is what we now call a $(C)$-point (C for Cauchy). Given a smooth function $f$ (defined on an open interval), a point $a$ in its domain is called a $(C)$-point iff the Taylor series of $f$ about $a$ converges in a positive interval, but the (analytic) function it defines does not coincide with $f$ in any interval centered at $a$.

The other way a smooth function $f$ may fail to be analytic at a point $a$ is if the Taylor series at $a$ has radius of convergence $0$. These points we now call $(P)$-points (P for Pringsheim). The first example is due to du Bois-Raymond in 1876. His example was the function $$ \sum_{n=0}^\infty (-1)^{n+1} \frac{x^{2n}}{(2n)! (x^2 + b_n)}, $$ where we start with a sequence $\{a_n\}$ of non-zero real numbers converging to $0$, and set $b_n = (a_n)^2$. (Again, the $(P)$-point is $0$ in this example.)

The classification into $(C)$- and $(P)$-points is due to Zahorski, who in 1947 published a paper characterizing these points. See here for a statement of Zahorski's result, and references.

Du Bois-Raymond's example was the basis for a theorem of Peano, from 1884, that nowadays we name after Borel (namely, the result that any sequence is the sequence of coefficients of the Taylor series about $0$ of a smooth function). Indeed, Peano's result went essentially unnoticed until very recently, when Besenyei publicized it. Borel's version of the result appeared in his thesis, in 1895. See here for some details and references.


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