Darboux Theorem: Advanced Analysis I

This lecture covers the Darboux theorem for continuous functions on closed intervals, stating that the difference between the upper and lower Darboux sums approaches zero as the partition becomes finer. The theorem is presented in various forms, emphasizing the uniform continuity of the function. The lecture also discusses the implications of the theorem for the continuity of functions and the behavior of function values on the interval. The proof techniques and key concepts related to the Darboux theorem are explored in detail, providing a deeper understanding of the properties of continuous functions on closed intervals.