Multivariable function equivalent definition

Question

Say a function $f: B(x;r) \rightarrow \mathbb{R}^q$ is a continuously differentiable function with $\|Jf(x)\| \leq c$ for all $x \in B(x;r)$. I want to show that $\|f(x_1) - f(x_2)\| \leq c\|x_1-x_2\|$ whenever $\|x_1-x_2\| < r$.

I have difficulty using the integration of the jacobian $Jf(x)$ to prove it, since it is not one-dimensional.

Answer 1

Since $f$ is $C^1$ you have $f(x_1)-f(x_2) = \int_0^1 Jf(x_2+t(x_1-x_2)) (x_1-x_2) dt$ and so $\|f(x_1)-f(x_2)\| \le (\int_0^1 c dt) \|x_1-x_2\|$.

The result is true if $f$ is just differentiable, but the proof needs a little more finesse.

Answer 2

This is wrong. Let $a>1$, and define $f:\>{\mathbb R}^2\to{\mathbb R}^2$ by $$f(x,y):=\left(ax,{y\over a}\right)\ .$$ Then $J_f\equiv1$, but $$\bigl\|f(x,0)-f(x',0)\bigr\|=a|x-x'|>1\cdot\bigl\|(x,0)-x',0)\bigr\|\ .$$