The Problem with Hilbert's Thirteenth Problem
... is that because Hilbert seemingly allowed any continuous funtions in the superposition sought, Kolmogorov and Arnold were able to show that yes indeed such a superposition can be constructed using highly non-differentiable functions. But the natural way to formulate the problem (as Hilbert implies in his later paper about the ninth degree equation) is to ask for a superposition using only algebraic functions. This problem is still unsolved!