This question was asked in an assignment that I am trying to solve.
Let $f:\mathbb{R} \to \mathbb{C}$ be continuous. Then prove the existence of an entire function $g$ such that $|f(x)-g(x)|< 1$ for all $x\in \mathbb{R}$.
I am sorry to say but for this question, I won't be able to provide any attempt as I don't know how to use the definition of continuity to prove the existence of $g.$
For background: have done a course in complex analysis, but that was marred due to a terrible instructor. I am not good at doing problems and hence trying.