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An elementary example is given by the formula $$f(x)=\sum_{k=1}^\infty k!\,\Phi(x-k!)$$ for real $x>1$, where $\Phi$ is the standard normal cumulative distribution function. Then the function $f$ is real analytic on $(1,\infty)$ (and can even be extended to an entire function) and $f(x)=O(x)$ for real $x>1$, whereas for natural $m$ we have $$\frac{f(m!)}{f(m!/2)}\sim\frac{m!/2}{(m-1)!}=\frac m2\to\infty$$ as $m\to\infty$.

An elementary example is given by the formula $$f(x)=\sum_{k=1}^\infty k!\,\Phi(x-k!)$$ for real $x>1$, where $\Phi$ is the standard normal cumulative distribution function. Then the function $f$ is real analytic on $(1,\infty)$ and $f(x)=O(x)$ for real $x>1$, whereas for natural $m$ we have $$\frac{f(m!)}{f(m!/2)}\sim\frac{m!/2}{(m-1)!}=\frac m2\to\infty$$ as $m\to\infty$.

The function $f$ can even be extended to the entire function $g$ defined by the formula $$g(z):=f(0)+\int_0^z dw \,\sum_{k=1}^\infty\frac{k!}{\sqrt{2\pi}}\,e^{-(w-k!)^2/2}$$ for complex $z$.

An elementary example is given by the formula $$f(x)=\sum_{k=1}^\infty k!\,\Phi(x-k!)$$ for real $x>1$, where $\Phi$ is the standard normal cumulative distribution function. Then $f$ is real analytic on $(1,\infty)$ and $f(x)=O(x)$ for real $x>1$, whereas for natural $m$ we have $$\frac{f(m!)}{f(m!/2)}\sim\frac{m!/2}{(m-1)!}=\frac m2\to\infty$$ as $m\to\infty$.

An elementary example is given by the formula $$f(x)=\sum_{k=1}^\infty k!\,\Phi(x-k!)$$ for real $x>1$, where $\Phi$ is the standard normal cumulative distribution function. Then the function $f$ is real analytic on $(1,\infty)$ (and can even be extended to an entire function) and $f(x)=O(x)$ for real $x>1$, whereas for natural $m$ we have $$\frac{f(m!)}{f(m!/2)}\sim\frac{m!/2}{(m-1)!}=\frac m2\to\infty$$ as $m\to\infty$.

An elementary example is given by the formula $$f(x)=\sum_{k=1}^\infty k!\,\Phi(x-k!)$$ for real $x>1$, where $\Phi$ is the standard normal cumulative distribution function. Then $f$ is real analytic on $(1,\infty)$ and $f(x)=O(x)$ for real $x>1$, whereas for natural $m$ we have $$\frac{f(m!)}{f(m!/2)}\sim\frac{m!/2}{(m-1)!}\sim\frac m2\to\infty$$ as $m\to\infty$.

An elementary example is given by the formula $$f(x)=\sum_{k=1}^\infty k!\,\Phi(x-k!)$$ for real $x>1$, where $\Phi$ is the standard normal cumulative distribution function. Then $f$ is real analytic on $(1,\infty)$ and $f(x)=O(x)$ for real $x>1$, whereas for natural $m$ we have $$\frac{f(m!)}{f(m!/2)}\sim\frac{m!/2}{(m-1)!}=\frac m2\to\infty$$ as $m\to\infty$.