Is it possible to construct an entire function that grows faster than any given continuous function.

Question

Assume that $f$ is a given real-valued continuous function over $\mathbb{R}$. How to construct an entire function $F$ such that $F(z)$ is real-valued when $z\in \mathbb{R}$ and there exists a real number $a$ such that $f(z)<F(z)$ for all $z>a,z\in \mathbb{R}$.