Feb 19, 2026

Multivariable analysis

Note. What I write here is mostly what I’ve learned from my amazing teacher, Katalin Nagy at BME. If something is incorrect here its my fault, not her’s.

Hello and welcome! We have arrived to another interesting math topic! Multivariable analysis. This is a really important field. It is used in physics a lot, and it is one of the key concepts to understand machine learning. So let’s roll up our sleeves and jump right in!

The topology of $\mathbb{R}^{p}$

👨‍🏫: Ok ok, in order to deeply understand this field we have to put down the basics and define some stuff. It will be a little boring, but it’s necessary to arrive to the interesting and beautiful part. So, let’s do that!

👨‍🏫: If we have a point in $\mathbb{R}^{p}$ we can denote it by:

x = (x_1, x_2, \dots, x_p)

So the point $x$ is a vector and $x_k$ is the $k$ -th component of the vector. Lucky for us we can measure distances in $\mathbb{R}^{p}$ and it’s really similar to how we measure distance in a plane (in $\mathbb{R}^{2}$ ).

If we are in ℝᵖ what could be the formula for measuring the length of a vector?

👨‍🏫: Yeah so it’s resembles to the Pythagorean theorem, but in higher dimension.

🙋‍♂️: But why does the “normal” square root method work, why don’t we use $p$ -th root?

\sqrt[p]{x_1^{p} + x_2^{p} + \dots + x_p^{p}}

👨‍🏫: You can think of this as the following: First, you only care about the first two components $x_1, x_2$ . You can apply the Pythagorean theorem (if you work in an orthonormal basis). After this, you take the result and the new component $x_3$ and do the same. If you repeat this until $x_p$ you’ll get this formula: $\sqrt{x_1^{2} + x_2^{2} + \dots + x_p^{2}}$ and the result is the length of $x$ which we denote: $\|x\|$ .

Norm (Euclidean)

For $x = (x_1, x_2, \dots, x_p) \in \mathbb{R}^{p}$ , the Euclidean norm of $x$ is:

\|x\| = \sqrt{x_1^{2} + x_2^{2} + \dots + x_p^{2}}

What other distances would be beneficial to use?

Open spheres

👨‍🏫: Good, we can move on and play with balls.

Definition: Open ball

Let $a \in \mathbb{R}^{p}$ and $\varepsilon > 0$ . The open ball of radius $\varepsilon$ centered at $a$ is:

B_{\varepsilon}(a) = \left\{ x \in \mathbb{R}^{p} : \|x - a\| < \varepsilon \right\}

👨‍🏫: What could this definition mean?

🙋‍♂️: Hmm… I guess $B_{\varepsilon}(a)$ is the set of points in $\mathbb{R}^{p}$ that lie within the sphere of radius $\varepsilon$ centered around $a$ .

👨‍🏫: Good! We call this an open ball because the points that are at a distance of $\varepsilon$ from $a$ are not in $B_{\varepsilon}(a)$ . You can think of it like a peeled orange.

Interior, exterior, boundary

👨‍🏫: Let’s look at this potato below. What could its interior, exterior, and boundary be?

🙋‍♂️: Ok. Now you intuitively understand these concepts. Let’s give their definition now.

Definition: Interior, Exterior, and Boundary Points

Let $A \subseteq \mathbb{R}^{p}$ and $x \in \mathbb{R}^{p}$ . We say $x$ is an…

① Interior point of $A$ , if:

\exists\, \varepsilon > 0 : B_{\varepsilon}(x) \subseteq A

② Exterior point of $A$ , if:

\exists\, \varepsilon > 0 : B_{\varepsilon}(x) \subseteq A^{c}, \quad \text{where } A^{c} = \mathbb{R}^{p} \setminus A

③ Boundary point of $A$ , if:

\forall\, \varepsilon > 0 : B_{\varepsilon}(x) \cap A \neq \emptyset \quad \text{and} \quad B_{\varepsilon}(x) \cap A^{c} \neq \emptyset

The set of all interior / exterior / boundary points of $A$ is called the interior / exterior / boundary of $A$ , denoted:

$\operatorname{int} A$ — interior
$\operatorname{ext} A$ — exterior
$\partial A$ — boundary

Examples

👨‍🏫: Let’s look at some examples now. What’s the interior, exterior, boundary of:

A = [5, 7[

B = \left\{ (x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} \leq 1 \right\}

Closure

👨‍🏫: Ok, what if we take a set $A$ and we want to include the boundary of $A$ too. Do we have a definition/notation for this too?

🙋‍♂️: Yes we do. We call it the closure of $A$ . And as you said it’s the set and the boundary points (we denote it by $\overline{A}$ ):

Definition: Closure

The closure of $A \subseteq \mathbb{R}^{p}$ is:

\overline{A} = A \cup \partial A

👨‍🏫: What is $\overline{A}$ and $\overline{B}$ in the problem above?

Open and closed sets

👨‍🏫: Now that we have closure let’s define open and closed sets.

Definition: Open and Closed Sets

Let $A \subseteq \mathbb{R}^{p}$ .

1. $A$ is open, if every point of $A$ is an interior point:

\forall x \in A : x \in \operatorname{int} A

2. $A$ is closed, if its complement is open:

A^{c} \text{ is open}

Two equivalent characterizations of closed sets:

If $A$ is closed, then $\partial A \subseteq A$
$\overline{A}$ is always a closed set

Note: Not every set is open or closed. The easiest counterexample is $[0, 1[$ .

👨‍🏫: Let’s unpack open sets a bit. Intuitively it means that however close we are to the boundary, we can always “draw” a sphere with radius $\varepsilon > 0$ that is entirely inside $A$ .

Properties of open and closed sets

👨‍🏫: We have two important statements for open sets and their pairs for closed sets.

Properties of Open and Closed Sets

Open sets:

(a) The union of arbitrarily many (even infinitely many) open sets is open.

(b) The intersection of finitely many open sets is open.

Closed sets:

(c) The intersection of arbitrarily many (even infinitely many) closed sets is closed.

(d) The union of finitely many closed sets is closed.

Connected sets

🙋‍♂️: Do we care about if a set is connected or not? Do we have a definition for that?

👨‍🏫: Yesss that’s an important idea too.

Definition: Connected Set

$A \subseteq \mathbb{R}^{p}$ is a connected set, if there do not exist open sets $C, B \subseteq \mathbb{R}^{p}$ such that:

C \cap B = \emptyset, \quad A \subseteq C \cup B, \quad A \cap C \neq \emptyset, \quad A \cap B \neq \emptyset

In other words: $A$ cannot be split into two non-empty, disjoint open parts.

Definition: Isolated Point

$a \in A$ (where $A \subseteq \mathbb{R}^{p}$ ) is an isolated point of $A$ if:

\exists\, \varepsilon > 0 : B_{\varepsilon}(a) \cap A = \{a\}

Limits and continuity

👨‍🏫: When we worked with regular functions we spent a lot of time analyzing limits and identifying whether a function is continuous or not. Will we do the same here too?

🙋‍♂️: Yeah, but first introduce some definitions!

Definition: Accumulation Point

$a \in \mathbb{R}^{p}$ is an accumulation point of $A \subseteq \mathbb{R}^{p}$ , if:

\forall\, \varepsilon > 0 \quad \text{there are infinitely many points of } A \text{ inside } B_{\varepsilon}(a)

Definition: Limit of a Sequence in ℝᵖ

$x \in \mathbb{R}^{p}$ is the limit of the sequence $(x_n) \subset \mathbb{R}^{p}$ , if:

\forall\, \varepsilon > 0 \quad \exists\, N = N(\varepsilon) : \forall n > N \quad x_n \in B_{\varepsilon}(x)

👨‍🏫: Some notes to these definitions. First we had another def for limits, it’s true here as well. Can you figure it out? Second, if the sequences are convergent by coordinates ( $\forall x_k \text{ is convergent}$ ) then $x_n$ is convergent as well. Third the Bolzano-Weierstrass theorem holds here as well. Fourth $\mathbb{R}^{p}$ is a complete space because every Cauchy-sequence is convergent.

Two variable functions

👨‍🏫: So if we have a function that “eats” two things and spits out one thing then we have a two variable function: $f: \mathbb{R}^{2} \to \mathbb{R}$ .

Definitions: Two-Variable Function

Let $f: D_f \subseteq \mathbb{R}^{2} \to \mathbb{R}$ and $C \in \mathbb{R}$ .

Graph:

\operatorname{graph}(f) = \left\{ (x, y, f(x, y)) : (x, y) \in D_f \right\}

Level curves:

\gamma_c = \left\{ (x, y) : f(x, y) = C, \quad (x, y) \in D_f \right\}

Contour lines:

\Gamma_c = \left\{ (x, y, f(x, y)) : f(x, y) = C, \quad (x, y) \in D_f \right\}

👨‍🏫: Graph of $f$ is straightforward. We’ll see many of them, they look like sheets. $\gamma_c$ are the contour lines. The ones that we see on detailed maps. And $\Gamma_c$ are the center lines in “their 3D place”.

Limit Continuity

Let’s look define limit and continuity for multivariable functions. Actually it will be really similar to what we did in single variable functions.

<AxiomBox lang="en" title="Definition: Limit of Multivariable Function">
Let f: \mathbb{R}^{p} \to \mathbb{R} let x, x_0 \in \mathbb{R}^{p}


</AxiomBox>