Example of continuous function which is not locally lipschitz

Question

A lot of analysis theorems admit for hypothesis locally lipschitz functions. I am wondering how to imagine those functions except as continuous ones. I can only see the Weierstrass function or other "fractal" functions as examples, but is there any simpler example (even in higher dimensional spaces, with more complicated metrics) that could be found in real problems (optimization, statistics, or PDE for instance) ?

Answer 1

Take$$\begin{array}{ccc}[0,+\infty)&\longrightarrow&\mathbb R\\x&\mapsto&\sqrt x,\end{array}$$for instance. It is not locally Lipschitz at $0$.