If $f:R\rightarrow R$ is a continuous function with the properties: $f(2017)=2016$ and $f(x)\cdot f(f(x))=1$ find the value of $f(2015)$
I replaced $x$ with $2017$ and got $f(2017)\cdot f(f(2017))=1$
$2016\cdot f(2016)=1$ which means $f(2016)=\frac1{2016}$, but I don't know how to get to $f(2015)$
You have that for every $y$ in the image of $f$ $f(y)=\frac 1y$. Furthermore 2016 and $\frac 1{2016}$ stay in the image, so since $f$ is continous also 2015 stays in the image. It follows that $f(2015)=\frac 1{2015}$
$f(t)=\frac 1 t$ for any $t$ in the range of $f$. Since $\frac 1 {2016}$ and $2016$ are both in the range and $f$ is continuous it any number between these two is in the range. So $f(2015)=\frac 1 {2015}$.
