Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 1 / 11E / c / Solution

No. The Darboux theorem says that every derivative has the intermediate-value property. Choose $ϵ<1/3$. In a deleted neighborhood of $a$, the assumed limit puts $\varphi (x)$ within $ϵ$ of $\varphi (a)+1$. Darboux's theorem applied between $a$ and any such $x$ would require values near $\varphi (a)+1/2$, which that neighborhood excludes. This contradiction shows that $\varphi$ cannot be a derivative.