For example, I could search for functions that are continuous, but that don't have differentiability, and come up with a continuous non-differentiable function. Or a smooth but non-analytical function. Or an commutative monoid without inverses. Or a compact topological space that is countable.
In cases where the object has been proven or conjectured to not exist, it could bring up the theorem or conjecture that states so. For example, if I search for differentiable non-continuous functions, it would bring up this theorem perhaps.
Does such a thing (or something close to it) exist?