# In the change-of-variables theorem, must $ϕ$ be globally injective?

In the above theorem, doesn't $\phi$ need to be injective too? The inverse function theorem merely implies that $\phi$ is locally injective -- is this sufficient? I ask because Marsden, in his Elementary Classical Analysis, actually does stipulate the condition that $\phi$ be (globally) injective.

p.s. In the above, "a set has volume" is equivalent to "the boundary of the set has measure zero."

Yes, the theorem you quoted is missing an injectivity assumption. It is possible to change variables by a non-injective map, but then one must account for multiple preimages in the integral on the right (multiply $f(x)$ by the number of preimages of $x$ under $\phi$).
As is, the following is a counterexample: let $n=2$, let $A$ be the annulus described in polar coordinates $(r,\theta)$ as $1<r<2$, and let $\phi(r,\theta)=(r,2\theta)$ be the "winding" map. Then $\phi(A)=A$, and $J\phi\equiv 2$. We can set $f\equiv 1$ and obtain a contradiction: $$\int_A 2 = \int_A 1$$