I am working on a proof for my real analysis class, and got stuck.
Let $U$ be an open subset of $\mathbb{R}^n$. Let $f: U \to \mathbb{R}^m$ be a continuously differentiable map, and further suppose that $Df(x_0)$ is injective for some $x_0 \in U$.
Show that there exists an open set $U_1 \subset U$ containing $x_0$ such that $f$ restricted to $U_1$ is injective.
I have already shown that for $x, y, x_0 \in U$, we have that $||f(x) – f(y) – Df(x_0)(x-y)|| \leq ||x-y||\sup||Df(v)-Df(x_0)||$ (*) by using the mean value inequality. Here I am taking the supremum over $v$, where $v$ is in the line segment connecting $x$ and $y$.
I was given a hint to apply this inequality to show that for some open ball $B_r (x_0)$, we have that $||f(x_1) – f(x_2)|| \geq C||x_1 – x_2||$ (**) for some constant $C$.
It is clear to me that injectivity follows from the second inequality (**), but I am struggling to show that this inequality is true.
The injectivity of $Df(x_0)$ gives us that $Df(x_0)(x-y) \neq 0$ for $x \neq y$, and I tried using this fact to break up the left hand side of (*) by, e.g., using the triangle inequality, but was unable to make much progress.
Best Answer
Consider $$c=\inf_{|v|=1} |Df(x_0)v|.$$ Since $L$ is injective one has $c>0$.
Now from (*), in a neighborhood of $x_0$ where $||Df(x)-Df(x_0)||<c/2$
you get $$ |f(x)-f(y)|\ge |Df(x_0)(x-y)| - \frac{c}{2}|x-y| \ge \frac{c}{2}|x-y|. $$