Is it possible to find two differentiable functions $f$ and $g$ and $g$ for which $x = f(x)g(x)$ and $f(0) = g(0) = 0$?
The fact that both functions have to be differentiable makes this a little more complicated, but we can say $\lim_{x \to a} \dfrac{f(x)-f(a)}{x-a} = \lim_{x \to a} \dfrac{\frac{x}{g(x)}-\frac{a}{g(a)}}{x-a}$ which has to be defined everywhere. Similarly with $g$. How can I prove this is possible?
Supposing that such functions exist, differentiating the equality yields $$\forall x, 1=f'(x)g(x) + g'(x)f(x)$$
Plugging $x=0$ yields $1=0+0$, a contradiction.
You cannot even have one of them being differentiable (the other one just continuous). Indeed, assume $f$ is differentiable. Then $$ f'(0) = \lim_{x\to 0}\frac{f(x)-f(0)}{x-0} = \lim_{x\to 0}\frac{f(x)}{x} = \lim_{x\to 0}\frac{1}{g(x)} = \infty. $$ Contradiction.
