We construct a continuous strictly increasing function such that at each point one of its right Dini derivates is $0$ or $\infty$, and at each point one of its left Dini derivates is $0$ or $\infty$. Thus at no point can it have a positive real unilateral derivative.
"On the Dini Derivates of a Particular Function." Real Anal. Exchange 25 (2) 943 - 946, 1999/2000.