I'm stuck in the following exercise:
Let $f:B_r (x_0) \subset \mathbb{R}^n \to \mathbb{R}^n$ be a diffeomorphism from $B_r (x_0)$ onto $f(B_r (x_0))$. If $\|f'(x)^{-1}\|\leq M$ for all $x\in B_r (x_0)$ and $|f(x_0)|\leq r/M$, then $0\in f(B_r (x_0))$.
I've tried using the mean-value inequality to find a sequence converging to zero, but it doesn't seem to be any hypothesis on the bound $M$, so I'm not sure if that's the right way to prove it. Since the only point of the domain I know something about is the center $x_0$, I guess I should try to show the origin belongs to some neighborhood of $f(x_0)$, but I'm not sure how to do it. Could someone help me?
Sketch: The derivative hypothesis $\|f'(x)^{-1}\| = \|(f^{-1})'(f(x))\| \leq M$ gives $\|f'(x)\| \geq 1/M$ for all $x$ with $\|x - x_{0}\| < r$. By the inverse function theorem, $f$ is a local diffeomorphism.
The mean value theorem shows that if $x$ and $y$ are arbitrary points of $B_{r}(x_{0})$, then $\|f(x) - f(y)\| \geq \frac{1}{M}\|x - y\|$. Particularly, if $0 < r_{0} < r$, then $f$ maps the sphere of radius $r_{0}$ about $x_{0}$ to the exterior of the ball of radius $r_{0}/M$ about $f(x_{0})$.
This suffices to imply the image $f(B_{r}(x_{0}))$ contains the ball $B_{r/M}(f(x_{0}))$, though you need topological machinery to prove this if $n > 1$. (On the assumption this is homework from topology or analysis, have you recently learned any such as relative homology or de Rham cohomology for detecting mapping degrees?)