Just for fun, I am looking for some interesting examples of functions that are uniformly continuous but not Lipschitz. I know the classic examples amongst functions from subsets of $\mathbb{R}$ to $\mathbb{R}$ (i.e square root and $x\sin(1/x)$).
Would love to see some examples of maps between more interesting spaces (so maybe I would put an emphasis on spaces that are not $\mathbb{R}^n$), although also open to learning some really cool/non-intuitive examples in $\mathbb{R}^n$.
If the proof that the function in question is uniformly continuous but not Lipschitz is not particularly obvious, I would also appreciate a nudge in the right direction or some guidance towards it.
Thank you!