Real Functions: Continuity Extension

This lecture covers the concept of extending a uniformly continuous function on a non-empty subset to a unique continuous function on the entire set. The proof involves demonstrating the existence of a limit independent of the chosen sequence, ensuring the continuity of the extension. The lecture also discusses the properties of Cauchy sequences and their convergence. The instructor illustrates how to define and prove the continuity extension, emphasizing the importance of uniform continuity. The presentation concludes by defining the continuous extension of a function and its significance in real functions.