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jstor.org/stable/10.4169/amer.math.monthly.121.01.060
Also, Kim and Kwon $[\mathbf{1 4}, \mathbf{1 7}]$ constructed examples of nowhere analytic increasing smooth functions in $\mathbb{R}$. An example of this kind is, for instance, the function $\int \psi(x) d x$, where $\psi: \mathbb{R} \rightarrow \mathbb{R}$ (see Figure 1) is given by
$$
\psi(x)=\sum_{j=1}^{\infty} \frac{1}{j !} \phi\left(2^{j} x-\left[2^{j} x\right]\right)
$$
where $[·]$ denotes the greatest integer function, and $\phi: \mathbb{R} \rightarrow \mathbb{R}$ is given by
$$
\phi(x)= \begin{cases}e^{-1 / x^{2}} \cdot e^{\frac{-1}{(x-1)^{2}}} & \text { if } 0<x<1 \\ 0 & \text { elsewhere }\end{cases}
$$
⋮
⋮
⋮
⋮ Let us now consider the minimum algebra of $\mathcal{C}(\mathbb{R})$ that contains the family of functions $\left\{\rho_{\alpha}\right\}_{\alpha \in \mathcal{H}}$ with $\rho_{\alpha}: \mathbb{R} \rightarrow \mathbb{R}$, defined as follows:
$$
\rho_{\alpha}(x)=\sum_{j=1}^{\infty} \frac{\lambda_{j}(x)}{\mu_{j}} \phi\left(2^{j} x-\left[2^{j} x\right]\right) \alpha^{j}
$$
As before, $[·]$ denotes the greatest integer function for $j \in \mathbb{N}$,
$$
\lambda_{j}(x)= \begin{cases}1 & \text { if }|x| \geq \frac{1}{2^{j}} \\ 0 & \text { otherwise }\end{cases}
$$
and $\mu_{j}=s_{k} !$ if $s_{k-1}<j \leq s_{k}$, where $s_{k}$ is the sum of the first $k$ positive integers.
Proposition 2.2. All functions $\left\{\rho_{\alpha}\right\}_{\alpha \in \mathcal{H}}$ are $\mathcal{C}^{\infty}$ and nowhere analytic. Moreover, all the derivatives and the function itself vanish at the points
$$
\left\{\frac{1}{2^{j}}: j \in \mathbb{N} \cup\{0\}\right\}
$$
Proof. The function $\phi(x)$ is smooth everywhere and analytic, except at $x=0$ and $x=1$. Moreover, it is flat at both of these points; that is, all its derivatives and the function $\phi(x)$ itself evaluated at those points are also 0 . If we replace $x$ by $2^{j} x-\left[2^{j} x\right]$, then the behavior of $\phi(x)$ over the interval $[0,1]$ is replicated by $\phi\left(2^{j} x-\left[2^{j} x\right]\right)$ on any dyadic interval of the form $\left[\frac{m-1}{2^{j}}, \frac{m}{2^{j}}\right]$ for all $m \in \mathbb{Z}$.
Let $\phi_{j}(x):=\frac{\lambda_{j}(x)}{\mu_{j}} \phi\left(2^{j} x-\left[2^{j} x\right]\right) \alpha^{j}$. Notice that, due to the flatness of $\phi$ at 0 and $1, \phi_{j}$ is smooth everywhere (and analytic everywhere but at $x=\frac{m}{2^{l}}$ for all $m \in \mathbb{Z}$ and $0 \leq l \leq j$ ). Now, $\sum_{j=1}^{\infty} \phi_{j}^{(k)}$ is uniformly bounded for all $k \in \mathbb{N}$, from which (by Remark 2.1) it follows that $\rho_{\alpha}$ is smooth for all $\alpha$ 's considered.
In order to prove that $\rho_{\alpha}$ is nowhere analytic, we follow the same procedure used by Kim and Kwon in [17, Theorem 1]. Assume that $\rho_{\alpha}$ is analytic at a point, so it is also analytic on an interval. Since the dyadics are dense, $\rho_{\alpha}$ is analytic at some $x_{0}=\frac{m}{2^{n}}$, with $m$ odd. If $1 \leq j \leq n-1$, then $\phi_{j}$ is analytic at $x_{0}$ and hence
$$
\widehat{\rho}_{\alpha}(x):=\sum_{j=n}^{\infty} \phi_{j}(x)
$$
is also analytic at $x_{0}$. However, $\widehat{\rho}_{\alpha}^{(k)}\left(x_{0}\right)=0$ for all integers $k \geq 0$. This contradicts the fact that $\widehat{\rho}_{\alpha}(x)$ is positive in some punctured neighborhood of $x_{0}$.
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