What was done in one variable calculus for scalar functions is generalized here to include the case of a vector valued function of possibly many variables. This part of the book is on functions of a single variable. However, it is no harder to consider the limit and continuity in terms of a function of many variables and it seems a good idea to go ahead and do it.
Definition 15.6.1 A function f : D
it follows that
f is continuous if it is continuous at every point of D
Note the total similarity to the scalar valued case. Also one obtains a similar description in terms of convergent sequences.
A repeat of the earlier theorem for functions of one variable yields the following equivalent description of continuity. All you have to do is make things bold face and repeat the earlier argument.