# Does for every continuous function $f:R \rightarrow R$ there exist a sequence of analytic functions convergent uniformly to $f$?

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is called $\mathbb{R}$-analytic iff for every $x_0 \in \mathbb{R}$ there exist $R>0$ and power series $\sum_{n=0}^\infty a_n (x-x_0)^n$ convergent for $|x-x_0|<R$ and such that $f(x)=\sum_{n=0}^\infty a_n (x-x_0)^n$ for $|x-x_0|<R$.

For some strange continuous functions on $\mathbb{R}$, for example for Weierstrass continuous nondifferentiable function (i.e. $f(x)=\sum_{n=0}^\infty a^n \cos(b^n \pi x)$ for $x\in \mathbb{R}$, where $0<a<1$, $b$ is positive odd integer such that $ab> 1+\frac{3}{2}\pi$), there exist a sequence of $\mathbb{R}$-analytic functions (even entire functions) which converges to $f$ uniformly (in the case of Weierstrass function it is sufficient to take the sequence of partial sum of series defining this function).

Is it maybe true that for every continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ there exists a sequence of $\mathbb{R}$ analytic functions which converges uniformly to $f$ ?

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Did you try for functions like $f(x)=\begin{cases}e^{-\frac 1{x^2}}&,x\neq 0\\\ 0&,x=0\end{cases}$? – Davide Giraudo Feb 11 '12 at 14:58
I believe this is true, yes. – Henno Brandsma Feb 11 '12 at 15:02
@Davide: How about $f_n(x)=\exp(-\frac{1}{x^2+1/n})$? – Henning Makholm Feb 11 '12 at 15:45
I'm not certain but I think a positive answer to the question is implied by Lemma 6 of "Analytic extensions of differentiable functions defined in closed sets", H Whitney, Trans. Amer. Math. Soc, 1934. – Colin McQuillan Feb 11 '12 at 15:49
@Davide: you are mixing "approximable by polynomials" with "approximable by Taylor polynomials". As I learned not that many years ago, it is often possible to find polynomials that approximate your function better than the Taylor polynomials. – Martin Argerami Feb 11 '12 at 18:24

a PDF of which is available (in French) from this page. In fact, he proves a bit more. A plain explanation can be found in other papers (e.g., here; I would post more links if I had more reputation). What I gather that he proved is, given a continuous function $f$ and a continuous notion of acceptable "error" (with error $\epsilon$ coming from choosing the constant function $\epsilon$), one can find an entire function approximating to the given precision.