Is it possible to have a $\textbf{monotonic}$ (i.e., non-increasing or non-decreasing) function $f : [a,b] \rightarrow \mathbb{R}$ $(a,b \in \mathbb{R}, a<b)$ which is differentiable on $[a,b]$, but has $\textbf{unbounded derivative}$, or equivalently, which is $\textbf{not Lipschitz continuous}$?
Are derivative functions always bounded on closed intervals? This has an answer, but without the condition that $f$ is monotonic.