continuously differentiable and local contraction

Question

Let $F$ be a map from $\mathbb{R}^n$ to $\mathbb{R}^n$. Fix $x_0\in \mathbb{R}^n$.

If $F$ is continuously differentiable near $x_0$ and the spectral radius of the Jacobian of $F$ at $x_0$ is less than 1, then I know that $F$ is a local contraction near $x_0$.

But, is the assumption that $F$ being continuosly differentiable "essential" for $F$ to be a local contraction near $x_0$?

For example, if I remove the assumption that $\mathbf{T}$ is continuously differentiable, but still assume that the spectral radius of the Jacobian of $F$ at $x_0$ is less than 1, is it possible for $T$ to fail to be a local contraction near $x_0?$

Answer 1

A local contraction is just a function that is Lipschitz around a point with Lipschitz constant less than one. Differentiability is not required.