Continuity and Differentiability of Functions of a Vector Variable

In the previous lesson the definition of the limit for a function f of the n variables \(x_{1}\) , \(x_{2}\) , \(\ldots \) , \(x_{n}\) (for short, they can be considered as the components of the vector \(\boldsymbol{x} \in \mathbb {R}^n\) ) has been given. In particular, the limit for a divergent function has been introduced. Examples of limits of such kind are discussed in the following. Moreover, some topological consequences of the continuity and the differentiability of functions of several scalar variables are presented.