An effective method of chaotification via time-delay feedback for a simple finite-dimensional continuous-time autonomous system is made rigorous in this paper. Some mathematical conditions are derived under which a nonchaotic system can be controlled to become chaotic, where the chaos so generated is in a rigorous mathematical sense of Li-Yorke in terms of the Marotto theorem. Numerical simulations are given to verify the theoretical analysis. Chaos has been found useful lately in various areas of science, engineering, and technology. Therefore, purposefully generating chaos (called chaotification, or anticontrol of chaos) has investigated rather intensively in the past few years. Recently, Wang, Chen, and Yu [Chaos 10, 771-779 (2000)] developed an anticontrol method via time-delay feedback for chaotifying a continuous-time dynamical system. The fundamental idea of this anticontrol method is correct and insightful, but a time-delay differential equation used therein is only approximated by a related discrete map, leaving some room for improvement. To present a mathematically rigorous approach, this paper adopts the same anticontrol idea but further improves its technical contents thereby deriving a similar yet rigorous design method for chaotificaiton. A rather general continuous-time system can be driven from nonchaotic to chaotic by using time-delay feedback perturbation on the system parameters or employing an exogenous time-delay state-feedback input, where the generated chaos is in a precise mathematical sense of Li-Yorke in terms of the Marotto theorem.