3 Tex corrected

As a matter of fact, real entire functions (that is, entire functions that map the real line into itself, or equivalently, functions represented by a power series centered in 0, with real coefficients and radius of convergence infinite) are dense in $C^0({\mathbb R}, \mathbb{R})$ in the sense of the order, that is:

Theorem (T.Carleman, 1927). For any two continuous real valued functions f < g there exists a real entire function $\phi$ in between:

$f(x)<\phi(x) < g(x)$ for all $x\in\mathbb{R}$.

So in particular, an entire function may be asymptotic to any continuous real function, and also, it may grow as fast as any continuous function.

2 minor

As a matter of fact, real entire functions (that is, entire functions that map the real line into itself, or equivalently, functions represented by a power series centered in 0, with real coefficients and radius of convergence infinite) are dense in $C^0({\mathbb R}, \mathbb{R})$ in the sense of the order, that is:

Theorem (T.Carleman, 1927). For any two continuous real valued functions f < g there exists a real entire function $\phi$ in between:

$f(x)<\phi(x) So in particular, an entire function may be asymptotic to any continuous real function, and also, it may grow as fast as any continuous function. 1 As a matter of fact, real entire functions (that is, entire functions that map the real line into itself, or equivalently, functions represented by a power series with real coefficients and radius of convergence infinite) are dense in$C^0({\mathbb R}, \mathbb{R})$in the sense of the order, that is: Theorem (T.Carleman, 1927). For any two continuous real valued functions f < g there exists a real entire function$\phi$in between:$f(x)<\phi(x)

So in particular, an entire function may be asymptotic to any continuous real function, and also, it may grow as fast as any continuous function.