We all know that if a function is continuously differentiable on a compact set $\Omega\subseteq \mathbb{R}^2$ then it is also Lipschitz continuous on that domain.
But now consider a function $f:\Omega \to \mathbb{R}$ that is uniformly continuous on but only piecewise continuously differentiable on the compact set $\Omega\subseteq \mathbb{R}^2$.
Can we say that such a function is Lipschitz continuous on $\Omega$? Why?
On
On the other hand, if $f'(x)=g(x)$ is piecewise continuous and uniformly bounded by some constant $M$ across each discontinuity then the theory of Riemann integration provides enough information to prove that $f(x)$ is locally Lipschitz. You will need to use the Fundamental Theorem of Calculus for piecewise continuous integrands to fill in the details.
No we can't. $\sqrt{|x|}$ is a counterexample.