The usual story goes like this:



> 
> **Smooth picture (?):**
> 
> 
> For a smooth bijection $\phi: M \to N$ between $n$-manifolds the following
>  is true:
> 
> 
> 1. $\phi^{-1}$ is a local diffeomorphism a.e.
> 2. Given an open set $U \subset N$ and a form $\omega \in \Omega^k(U)$ we have the equality: $\int\_U \omega= \int\_{\phi^{-1}(U)} \phi^\*\omega$.
> 
> 
> 


Satisfied with this simple and very intuitive picture I slowly came to believe that this is the most general change of variables theorem there could be. That is, until I met the following theorem in a measure theoretic context.



> 
> **Theorem 1:**
> 
> 
> Let $U\subset\mathbb{R}^n$ be a measurable subset and $\phi :U\to \mathbb{R}^n$ be injective and differentiable. 
> 
> 
> $\implies$ $\int\_{\phi(U)} f = \int\_U f\circ\phi |\det D\phi|$ for all real valued functions $f$.
> 
> 
> 


Then upon searching the internet I came by a weaker version of the inverse function theorem for everywhere differentiable functions:



> 
> **Theorem 2:**
> 
> 
> Let $U \subset \mathbb{R}^n$ and $f:U \to \mathbb{R}^n$ be differentiable s.t. $Df\_{x\_0}$ has full rank for all $x\_0 \in U$  
> 
>  $\implies$ $f$ is a local differentiable homeomorphism.
> 
> 
> 


Which leads to the following generalization:



> 
> **Differentiable picture (conjecture):**
>  For a differentiable bijection $\phi: M \to N$ between $n$-manifolds the following
>  is true:
> 
> 
> 1. $\phi^{-1}$ is a local differentiable homeomorphism a.e.
> 2. Given an open set $U \subset N$ and a form $\omega \in \Omega^k(U)$ we have the equality: $\int\_U \omega= \int\_{\phi^{-1}(U)} \phi^\*\omega$.
> 
> 
> 


In search of a unifying picture i listed the properties a function $\phi$ must have so that the pullback wont change the value of the integral.


**I.** $\phi$ must be locally absolutely continuous. (otherwise it could send a null set to a positive measure set). This also establishes the almost everywhere differentiability of $\phi$.


**II.** $D \phi$ must have full rank almost everywhere.



> 
> If we can preform change of variables with $\phi$. What more properties must it have?
> 
> 
> 


Following the connection with lipschitz functions i arrived at the following unifying conjecture:



> 
> **The great conjecture:**
> 
> 
> For a locally lipschitz bijection $\phi: M \to N$ between $n$-manifolds the following is true:
> 
> 
> 1. $\phi^{-1}
> $ is locally bi-lipschitz (a.e.?)
> 2. Given an open set $U \subset N$ and a form $\omega \in \Omega^k(U)$ we have the equality: $\int\_U \omega= \int\_{\phi^{-1}(U)} \phi^\*\omega$.
> 
> 
> 


Is this true?



Integrals of pullbacks and the Inverse function theorem(s?)
If you consider continuous injections (resp. homeomorphisms onto their range) instead of locally Lipschitz bijections (resp. locally bi-Lipschitz), then the modified conjecture is true because of [Brouwer's theorem on invariance of domain](http://en.wikipedia.org/wiki/Invariance_of_domain), with the proviso that in (2) you should consider $n$-forms instead of $k$-forms because the domain $U$ of integration is assumed open and hence is $n$-dimensional. Formula (2) in this case is simply the [change of variables formula](http://en.wikipedia.org/wiki/Pushforward_measure) for integrals with respect to a measure $\mu$ and its pushforward $\phi\_\*\mu$, once you write $n$-forms in terms of the volume measure associated to a Riemannian metric on $M$ (recall that a continuous map is always Borel measurable). Here I'm assuming $M$ and $N$ oriented for simplicity.


Otherwise (assuming back your original hypotheses), you should consider [locally $k$-rectifiable subsets](http://en.wikipedia.org/wiki/Rectifiable_set) $U$ of the target manifold $N$ in (2) instead of $U$ open ("locally" equals "countably" if your manifolds are second countable). All that remains is to show that the inverse of $\phi$ is locally Lipschitz. This, however, needs an additional hypothesis as follows. By [Rademacher's theorem](http://en.wikipedia.org/wiki/Rademacher%27s_theorem), any locally Lipschitz map $\phi:M\rightarrow N$ is differentiable almost everywhere; more precisely, the (Clarke sub)differential $D\phi$ of $\phi$ is a locally bounded set-valued map which is single-valued almost everywhere. That being said, the additional hypothesis you need in your conjecture to settle part (1) is that $D\phi$ *takes only non-singular values*, for in this case you can invoke Clarke's inverse function theorem for Lipschitz maps (Theorem 1 of F.H. Clarke, "On The Inverse Function Theorem". [Pac. J. Math. **64** (1976) 97-102](http://projecteuclid.org/euclid.pjm/1102867214)). Part (2) follows immediately by using the $k$-dimensional Hausdorff measure with respect to some Riemannian metric on $M$.


Some of the comments addressed the possibility of doing away with the hypothesis of $D\phi$ being non-singular by invoking Sard's theorem. However, this does not work with so little regularity assumed from $\phi$ - a necessary hypothesis for Sard's theorem to hold true, as pointed in Sard's original paper ("The Measure of Critical Values of Differentiable Maps", [Bull. Amer. Math. Soc. **48** (1942) 883-890](http://www.ams.org/journals/bull/1942-48-12/S0002-9904-1942-07811-6/)) is that $\phi$ should be at least $\mathscr{C}^1$. $\phi$ being locally Lipschitz is not good enough - as shown by J. Borwein and X. Wang ("Lipschitz functions with maximal subdifferentials are generic", [Proc. Amer. Math. Soc. **128** (2000) 3221–3229](http://www.ams.org/journals/proc/2000-128-11/S0002-9939-00-05914-1/)), the set of 1-Lipschitz functions $f$ such that *all* points in its domain are critical (in the sense that $Df$ contains zero everywhere) is *generic* with respect to the uniform topology. In particular, generically one cannot expect the inverse of a locally Lipschitz bijection, albeit certainly continuous, to be locally Lipschitz (even if only up to a subset of measure zero).


