calculus - real analysis problem about $C^1$ functions

Let $f: I \to \mathbb{R}$ be a $C^1$ function, where $I$ is an open interval in $\mathbb{R}$. Suppose that $f(0)\not = 0$. Prove that there are neighborhoods $U$ and $V$ of $0$ in $\mathbb{R}$ such that for all $x\in U$ there exists a unique $y\in V$ such that $$xf(y) = \int_0^y f(xt) dt,$$ and prove that the function $g:x\in U \mapsto y \in V$ defined by this is a $C^1$ function and satisfies $g'(0) = 1$.

How do I prove this? In reading through it, it seems that it may involve the Mean Value Theorem, but I just cant seem to piece it together and don't really know where to start.

thanks to all who take time to help