Sets where \(\operatorname{Lip}f\) is infinite and \(\operatorname{lip}f\) is finite (Q2661208)

Let \(f:\mathbb{R}\to \mathbb{R}\). Denoting \(\operatorname{Lip}f= \lim\sup_{r\to 0+} M_{f}(x, r)\) and \(\operatorname{lip}f= \lim\inf_{r\to 0+} M_{f}(x, r)\), where \(M_{f}(x, r)=\sup \left\{\frac{f(x)-f(y)}{r} : |x-y|\le r \right\}\), in this paper the author characterizes the sets \(E \subset \mathbb{R}\) such that there exists a continuous function \(f :\mathbb{R}\to \mathbb{R}\) with \(\operatorname{lip}f\) finite everywhere and \(\operatorname{Lip}f\) infinite precisely on the set \(E\).