$C^1(\mathbb{R}^n)$ implies locally Lipschitz in $\mathbb{R}^n$ According to wikipedia a function $f\colon \mathbb{R}^n\to\mathbb{R}^n$ that is continuously differentiable, is also locally Lipschitz.
I there someone who knows a good reference which contains a proof of this statement?

The proof on $\mathbb{R}^n$ is fairly straightforward.
Choose some ball $B(\hat{x},\epsilon)$. The closure is compact, so the derivative $\frac{\partial f}{\partial x}$ is bounded by some $L$ on the ball. Now suppose $x,y \in B(\hat{x},\epsilon)$, then using Taylor's formula, we have:
$$f(x)-f(y) = \int_o^1 \frac{\partial f (y+t(x-y))}{\partial x}(x-y)\;dt.$$
Hence we can get the bound:
$$\|f(x)-f(y) \| \leq \int_o^1 \|\frac{\partial f (y+t(x-y))}{\partial x}\|\; \|(x-y)\| \;dt \leq L \|x-y\|.$$ 
