### Instruction: Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a function satisfying $f(x)f(y)=f(x-y)$. Find all possible values of $f(2017)$.  Enter all the possible values, separated by commas.### Response: This is a functional equation, and it might have multiple solutions.
I want to see if I can find some properties of $f$ that can help me narrow down the possibilities.
One thing I notice is that if I plug in $x=y$, I get $f(x)^2=f(0)$, so $f(x)$ is either $\sqrt{f(0)}$ or $-\sqrt{f(0)}$ for all $x$.
Another thing I notice is that if I plug in $y=0$, I get $f(x)f(0)=f(x)$, so either $f(0)=1$ or $f(x)=0$ for all $x$.
If $f(0)=1$, then $f(x)=\pm 1$ for all $x$, and $f(2017)=\pm 1$.
If $f(x)=0$ for all $x$, then $f(2017)=0$.
So the possible values of $f(2017)$ are $-1, 0, 1$.