A proof regarding Lebesgue measure and a differentiable function Could anyone kindly provide a hint on the following problem? My guess is to do some change of variables? Thank you!

Let $f:\Bbb{R}\rightarrow\Bbb{R}$ be a continuously differentiable function. Define $\phi:\Bbb{R}^2\rightarrow\Bbb{R}^2$ by $\phi(x,y)=(x+f(x+y),y-f(x+y))$. Prove that $|\phi(E)|=|E|$ for every measurable set $E\subset \Bbb{R}^2$, where $|E|$ denotes the Lebesgue measure of $E$.


Consider the change of variables
$$\begin{align}
\xi&=x+f(x+y,)\\ \eta &= y-f(x+y).
\end{align}$$
It is bijective, the inverse being
$$\begin{align}
x&=\xi-f(\xi+\eta),\\ y&= \eta+f(\xi+\eta).
\end{align}$$
Its jacobian is
$$
\frac{\partial(\xi,\eta)}{\partial(x,y)}=1.
$$ change of variables in multiple integrals gives the result.
The formula for
