Prove that $f(x)=x$.

Question

Let $f:\mathbb{R}\to\mathbb{R}$ be a function that satisfies

• $f(-x)=-f(x)$
• $f(x+1)=f(x)+1$
• $f\left(\frac1x\right)=\frac{f(x)}{x^2}$ for $x\neq0$.

Prove that $f(x)=x$.

I am even interested in the case where $f$ is continuous.

I found this question in my notebook from a while back and I hadn't written a solution then. I have no idea where to begin.

Any help would be appreciated!