This technical note presents a novel characterization of controllability for linear time-invariant finite-dimensional systems. This characterization relates eigenvalue controllability with the continuity of the map that assigns to each closed-loop eigenvalue the smallest subspace containing the set of corresponding closed-loop eigenvectors. Application of the given characterization is illustrated on a specific case of controller-driven sampling stabilization, where the sampled system is interpreted as a discrete-time switched system and stability under arbitrary switching is ensured via simultaneous triangularization (Lie-algebraic solvability).