A function $f: \mathbb{R} → \mathbb{R}$ is defined such that, $$f\left(\frac{x+y}{x-y}\right) = \frac{f(x) + f(y)}{f(x)-f(y)} $$ for all $x \neq y$. Prove that $f(x) = x$ for all $x \in \mathbb{R}$.
My progress on this problem was first showing $f(0)=0, f(1)=1, f(-1)=-1.$ Then by some algebraic manipulation, I showed $f(x)f(y) = f(xy)$ from where it follows that $f$ is an odd function. If $f$ were continuous, I'd be done by Cauchy FE, but unfortunately it isnt and I dont know where to proceed from here, any hints or techniques would be greatly appreciated.
The forum that I was linked to didn't have satisfactory answers. Some of them assumes $f$ is differentiable and I didnt really understand one of the solutions. The further progress that I got to was $x>y \iff f(x) > f(y)$ and thus we get $f$ is injective.