A simultaneous solution to two problems on derivatives - F. S. Cater (International Journal of Mathematics and Mathematical Sciences)
Let A be a nonvoid countable subset of the unit interval [0,1] and let B be an Fσ-subset of [0,1] disjoint from A. Then there exists a derivative f on [0,1] such that 0≤f≤1, f=0 on A, f>0 on B, and such that the extended real valued function 1/f is also a derivative.