The function $f(x)=(-1)^x$ can only be mapped to the real numbers when $x\in\mathbb{N}$. So I wonder what analytic continuation can expand the domain of $f$ to all real numbers without changing the range from $\mathbb{R}$ to $\mathbb{C}$. What continuous functions do this?
I found that $(-1)^x=\cos((2k+1)\pi x)$ when $x\in\mathbb{N},k\in\mathbb{Z}$ and the RHS is a function $\mathbb{R}\mapsto\mathbb{R}$. Are there any others?
Well, $(-1)^x = e^{(i\pi + 2\pi i k)x}$ for any integer $k$. So if you just take the real part, $\cos((2k + 1)\pi x) $, you get a "standard" family of real valued functions that agree with $(-1)^x$ whenever $x$ is an integer.
Aside from that as others have commented, you can just add any function which zero at all integers, beyond that there is nothing more interesting to say about the question.