Let me first give a definition of a diffeomorphism:
A map $f:\mathbb R^n\rightarrow \mathbb R^n$ is a diffeomorphism if it is invertible with inverse $g:=f^{-1}$, differentiable and with differentiable inverse.
(I know that often, $f$ and $g$ are required to be $C^{1}$, but I'd like to stick to only differentiability.)
Now, does every diffeomorphism $f$ fulfill $\det Jf(\mathbf{a}) \ne 0$ for all $a\in\mathbb R^n$, where $Jf$ denotes the Jacobi matrix of $f$? If not, are there any counterexamples? Personally, I've tried one-dimensional functions such as $f(x) = x^3+x$ or $f(x) = mx +b$, but they fulfill $\det Jf(\mathbf{a}) \ne 0\dots$ for all $a\in \mathbb R^{n}\dots$
Fix $a\in\Bbb R^n$ and put $b:=f(a)$.
If $g$ is an inverse of $f$ then $f\circ g=\mathrm{id}$, that is $g(f(x))=x$ for all $x\in\Bbb R^n$. They are both differentiable, so differentiating this equation and using the chain rule we get $Dg(b)\circ Df(a)=\mathrm{id}$. Thus $Jg(b)\cdot Jf(a)=\mathrm{I}$, where $\mathrm{I}$ is an identity matrix. This shows that $Jf(a)$ is invertible.