A uniqueness problem concerning entire functions and their derivatives

We determine all entire functions $f$ such that for nonzero complex values $a\mathrm{\ne }b$ the implications $f=a\Rightarrow f^{\mathrm{\prime }}=a$ and $f^{\mathrm{\prime }}=b\Rightarrow f=b$ hold. This solves an open problem in uniqueness theory. In this context we give a normality criterion, which might be interesting in its own right.