Functions of several variables

${\mathbb{R}}^n$ is the set of ordered $n$-tuples of real numbers $\mathbf{x}=(x_1,\dots ,x_n)$. For $n=2$ points are written $(x,y)$, for $n=3$ they are written $(x,y,z)$. Addition and multiplication by a scalar are carried out componentwise:

Functions of several variables are defined on subsets of ${\mathbb{R}}^n$; the definitions in this chapter are stated for arbitrary $n$, and in the exercises they are almost always used with $n=2$.

1. Euclidean norm and distance

The Euclidean norm of $\mathbf{x}\in {\mathbb{R}}^n$ is

The distance between two points $P$ and $Q$ is $d(P,Q)=\Vert P-Q\Vert$; in the plane, for $P=(x_1,y_1)$ and $Q=(x_2,y_2)$,

For $n=1$ the norm coincides with the absolute value, so the norm extends to ${\mathbb{R}}^n$ the tool with which closeness is measured in Mathematical Analysis 1: every condition of the form $|x-x_0|<\delta$ becomes $\Vert P-P_0\Vert <\delta$.

2. Circular neighborhood

For $P_0\in {\mathbb{R}}^n$ and $r>0$, the neighborhood of $P_0$ of radius $r$ is

In ${\mathbb{R}}^2$ it is the open disk with center $P_0$ and radius $r$, without its boundary circle; in ${\mathbb{R}}^3$ it is the open ball. The circular neighborhood replaces the interval $(x_0-r,x_0+r)$ of Mathematical Analysis 1 and appears in all the definitions that follow: interior points, open sets, limits and continuity.

1. Interior, exterior and boundary point

Let $A\subseteq {\mathbb{R}}^2$ and $P_0\in {\mathbb{R}}^2$. The point $P_0$ is

• interior to $A$ if there exists a neighborhood $B(P_0,r)$ contained in $A$;
• exterior to $A$ if there exists a neighborhood $B(P_0,r)$ contained in the complement of $A$;
• a boundary point of $A$ if every neighborhood of $P_0$ contains both points of $A$ and points of the complement.

Every point of the plane is of exactly one of the three types. The set of interior points is denoted by $\mathring{A}$, the set of boundary points by $\partial A$. The classification of points is used to recognize open and closed sets, which appear in the hypotheses of the theorems of this chapter.

2. Accumulation point

$P_0\in {\mathbb{R}}^2$ is an accumulation point of $A$ if every neighborhood of $P_0$ contains at least one point of $A$ different from $P_0$. The point may or may not belong to $A$. A point of $A$ that is not an accumulation point of $A$ is called isolated. As in one variable, the limit as $(x,y)\to (x_0,y_0)$ makes sense only if $(x_0,y_0)$ is an accumulation point of the domain: in the exercises the typical case is the origin, an accumulation point of ${\mathbb{R}}^2\setminus \{(0,0)\}$.

3. Open set and closed set

$A\subseteq {\mathbb{R}}^2$ is open if each of its points is interior, that is, if $A=\mathring{A}$; it is closed if its complement is open, a condition equivalent to $\partial A\subseteq A$. The disk $x^2+y^2<1$ is open; the disk $x^2+y^2\le 1$ is closed. In practice, a set defined by strict inequalities between continuous functions is open, while a set defined by non-strict inequalities or by equalities is closed; this criterion makes it possible to classify the domains found in the exercises.

4. Bounded set and compact set

$A\subseteq {\mathbb{R}}^2$ is bounded if it is contained in a disk centered at the origin, that is, if there exists $R>0$ with $\Vert P\Vert \le R$ for every $P\in A$. A closed and bounded set is called compact. The closed disk $x^2+y^2\le 4$ is compact; the half-plane $y\ge 0$ is closed but not bounded, so it is not compact. Compact sets are the sets on which the Weierstrass theorem in several variables applies, and problems of absolute maximum and minimum are set up on them.

5. Connected set

$A\subseteq {\mathbb{R}}^2$ is path-connected if every pair of points of $A$ can be joined by a continuous curve entirely contained in $A$. A disk and an annulus are connected; the union of two disjoint disks is not. On an open connected set a continuous function that vanishes at no point has constant sign, so connectedness enters sign studies: the sign is determined at a sample point for each connected piece.