Is $Isubseteqmathbb{R}$ an intervall and $f: Itomathbb{R}$ a differentiable function with bounded derivative $f’: Itomathbb{R}$, then $f$ is Lipschitz-continuous.

This is supposed to be an application of the mean-value theorem.

What gets me is the use of unspecified intervalls. So $I=(a,b), (a,b), (a,b), (a,b)$, as the mean-value theorem holds for differentiable functions defined on a compact intervall (a,b).

Every resource I looked it up proofs this result for compact intervalls, but I was unable to give a counterexample for say $I=(a,b)$, because of the bounded derivative.

But how does one relax the condition to $I=(a,b)$ to apply the mean-value theorem?

I thought that one might can proof that for $I=(a,b)$ you are able to continuously extend to $(a,b)$.

Thanks in advance.