Derivative test for contraction mapping on open sets I have a question on the derivative test for showing that a function is a contraction. The proposition that I know is in this version:

Let $I$ be a closed bounded set of $\mathbb{R}^l$ and $f : I → I$
  a differentiable function with $||f'(x)|| \leq \beta$, 
  $0<\beta<1$, for all $x \in I$. Then f is a contraction with module of
  contraction equal to $\beta$.

Does the following proposition hold too? (I have put in bold the changed parts)
Let $I$ be an open set of $\mathbb{R}^l$ and $f : I \rightarrow \pmb{\mathbb{R}^l}$ a
differentiable function with $||f'(x)|| \leq \beta$,  $0<\beta<1$, for all $x \in I$. Then f is a contraction on $I$ with module of contraction equal to $\beta$. 

This is not true. Consider the function that maps an open ball (say, the unit ball) to $c$ for some constant $c$
