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|
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, $b$ 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 $f$ ?