Lecture 1. Introduction to the Analysis of Functions of Several Variables

This lecture introduces functions of several variables, explaining the roles of independent and dependent variables, domain, and range. It covers methods for visualizing such functions, including graphs in three-dimensional space and level lines on a plane. The core topic is the limit of a function of two variables. For a limit to exist, it must be the same regardless of the path taken to approach the point. Examples demonstrate how different curves can yield different limit values, proving that a limit does not exist. The relationship between double and iterated limits is also discussed. Finally, the lecture addresses continuity, defining when a function is continuous at a point and classifying discontinuities. It concludes with the Weierstrass and Cauchy theorems, explaining why properties like closedness and connectedness are essential for continuous functions on sets.