# Find $f(2015)$ if f is continuous and $f(2017)=2016$ [duplicate]

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 $$f(2015)$$

• Feb 11 '20 at 9:56
• many thanks, you saved me Feb 11 '20 at 10:00

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 $$f(2015)=\frac 1{2015}$$
• If $\displaystyle f(y)=\frac1y$, how can we have $f(2017)=2016\not=\frac1{2017}$? Feb 11 '20 at 10:56
• @Dr.Mathva it means that 2017 doesn’t stay in the range of $f$! Feb 11 '20 at 10:59
$$f(t)=\frac 1 t$$ 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 $$f(2015)=\frac 1 {2015}$$.