If a function is Lipschitz, and differentiable, is its gradient also Lipschitz?

Question

If $f(x)$ is Lipschitz, i.e. $$||f(x) - f(y)|| \le L||x-y||$$ is it's gradient also Lipschitz?

$$||\nabla f(x) - \nabla f(y)|| \le K||x - y|| $$

And does $L = K$ ?

Answer 1

By considering a function which depends only on the first coordinate you can reduce this to a question on the real line.

Let $h$ be an intergable function on $\mathbb R$ which is not bounded. Let $g(x)=\int_0^{x} h(t)dt$ and $f(x)=\int_0^{x} g(t)dt$. Since $f$ has a bounded derivative (namely $g$) we see that $f$ is Lipschitz. However $f'=g$ is not Lipschitz: If it is Lipschitz then $h$ would be bounded since $g'=h$ almost everywhere.