Real Functions: Continuity and Limits

This lecture covers the concept of continuity for real functions, defining it as the existence of a limit at a point and the function value being equal at that point. Examples illustrate how functions can be discontinuous at certain points while remaining continuous when fixing other variables. The lecture also introduces the notion of uniform continuity, highlighting the difference between continuity and uniform continuity. Various theorems and criteria, such as the Cauchy criterion, are presented to determine the continuity and limits of functions. The equivalence between different definitions of continuity is discussed, emphasizing the importance of understanding the relationship between limits and continuity. The lecture concludes with remarks on the distinction between continuous and uniformly continuous functions.