Q1: Let $v: \mathbb{R} \rightarrow \mathbb{R}$ be a locally Lipschitz continuous function. What is not correct in general?

A1: $v$ is continuously differentiable.

A2: $v$ is continuous.

A3: For every point $x \in \mathbb{R}$ there is an $\varepsilon > 0$ and a constant $L > 0$ such that for all points $y,z \in (x- \varepsilon, x+\varepsilon)$, we have $$ |v(y) - v(z) | \leq L | y - z | $$

Q2: Let $v: \mathbb{R} \rightarrow \mathbb{R}$ be given by $v(x) = x^2$. Is $v$ locally Lipschitz continuous?

A1: Yes!

A2: No!

A3: One needs more information.

Q3: Let $v: \mathbb{R} \rightarrow \mathbb{R}$ be given by $v(x) = \sqrt{|x|}$. Is $v$ locally Lipschitz continuous?

A1: No!

A2: Yes

A3: One needs more information.