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\begin{document}


%----------------------------------------------------------------------------------------
%	TITLE PAGE
%----------------------------------------------------------------------------------------

\titlepage % Output the title page
	{\includegraphics[width=\paperwidth]{background.pdf}} % Code to output the background image, which should be the same dimensions as the paper to fill the page entirely; leave empty for no background image
	{ % Title(s) and author(s)
		\centering\sffamily % Font styling
		{\Huge\bfseries Mathematical Analysis Dictionary\par} % Book title
		\vspace{16pt} % Vertical whitespace
		{\LARGE Definitions, Theorems, and Essential Formulas\par} % Subtitle
		\vspace{24pt} % Vertical whitespace
		{\huge\bfseries Enrico Bertolazzi\par} % Author name
	}

%----------------------------------------------------------------------------------------
%	COPYRIGHT PAGE
%----------------------------------------------------------------------------------------

\thispagestyle{empty} % Suppress headers and footers on this page

~\vfill % Push the text down to the bottom of the page

\noindent Copyright \copyright\ 2026 Enrico
Bertolazzi\\ % Copyright notice

\noindent \textsc{University of Trento}\\ % Publisher

\noindent \textsc{\href{https://www.latextemplates.com/template/legrand-orange-book}{book-website.com}}\\ % URL

\noindent Licensed under the Creative Commons Attribution-NonCommercial 4.0 License (the ``License''). You may not use this file except in compliance with the License. You may obtain a copy of the License at \url{https://creativecommons.org/licenses/by-nc-sa/4.0}. Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an \textsc{``as is'' basis, without warranties or conditions of any kind}, either express or implied. See the License for the specific language governing permissions and limitations under the License.\\ % License information, replace this with your own license (if any)

\noindent \textit{English edition, 2026} % Printing/edition date

%----------------------------------------------------------------------------------------
%	TABLE OF CONTENTS
%----------------------------------------------------------------------------------------

\pagestyle{empty} % Disable headers and footers for the following pages

\tableofcontents % Output the table of contents

%\listoffigures % Output the list of figures, comment or remove this command if not required

%\listoftables % Output the list of tables, comment or remove this command if not required

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\cleardoublepage % Start the following content on a new page

\bookmarksetup{startatroot}

\chapter{Notation and Preliminary
Concepts}\label{notation-and-preliminary-concepts}

\index{notation and preliminary concepts@Notation and Preliminary Concepts}

\begin{tcolorbox}[enhanced jigsaw, arc=.35mm, bottomrule=.15mm, bottomtitle=1mm, breakable, colback=white, colbacktitle=quarto-callout-note-color!10!white, colframe=quarto-callout-note-color-frame, coltitle=black, left=2mm, leftrule=.75mm, opacityback=0, opacitybacktitle=0.6, rightrule=.15mm, title={Purpose and Scope}, titlerule=0mm, toprule=.15mm, toptitle=1mm]

This is not a set of lecture notes. It is a compact dictionary of
definitions and results used in the accompanying courses. If you want to
study any of these topics, you should follow a course or consult a
textbook devoted to the subject. This dictionary is intended for readers
who already know, or have previously studied, the material and need a
concise reference whenever a reminder is useful.

\end{tcolorbox}

In the following, unless otherwise stated,
\[f:A\subseteq\mathbb{R}^n\to\mathbb{R}^m.\] The Euclidean norm is
indicated with \(\|\cdot\|\).

For \(x_0\in\mathbb{R}^n\):

\[B_r(x_0)=\{x:\|x-x_0\|<r\},
\qquad
\overline B_r(x_0)=\{x:\|x-x_0\|\le r\}.\]

A point \(x_0\) is accumulation for \(A\) if every \(B_r(x_0)\) contains
points of \(A\) different from \(x_0\).

\section{Interior, Closure, and
Boundary}\label{interior-closure-and-boundary}

\index{interior closure and boundary@Interior, Closure, and Boundary}

\[\operatorname{int}(A)=\{x:\exists r>0,\ B_r(x)\subset A\},\]
\[\operatorname{cl}(A)=\text{smallest closed set containing }A,\]
\[\partial A=\operatorname{cl}(A)\setminus\operatorname{int}(A).\]

\bookmarksetup{startatroot}

\chapter{Sequences and Limits}\label{sequences-and-limits}

\index{sequences and limits@Sequences and Limits}

\section{Convergence of Sequences}\label{convergence-of-sequences}

\index{convergence of sequences@Convergence of Sequences}

A sequence \(x_k\in\mathbb{R}^n\) converging \(x\) if
\[\forall\varepsilon>0\ \exists N:\ k\ge N\Rightarrow \|x_k-x\|<\varepsilon.\]
It is written \[x_k\to x.\] In \(\mathbb{R}^n\):

\[x_k\to x
\iff
(x_k)_i\to x_i\quad\forall i.\]

\section{Cauchy Criterion}\label{cauchy-criterion}

\index{cauchy criterion@Cauchy Criterion}

In \(\mathbb{R}^n\):

\[x_k\text{ converges}
\iff
\forall\varepsilon>0\ \exists N:\ p,q\ge N\Rightarrow\|x_p-x_q\|<\varepsilon.\]

\section{Bolzano--Weierstrass Theorem}\label{bolzanoweierstrass-theorem}

\index{bolzano-weierstrass theorem@Bolzano–Weierstrass Theorem}

Every bounded sequence in \(\mathbb{R}^n\) admits a converging
subsequence.

\section{Limit of a Function}\label{limit-of-a-function}

\index{limit of a function@Limit of a Function}

We say \[\lim_{x\to x_0}f(x)=L\] if
\[\forall\varepsilon>0\ \exists\delta>0:
0<\|x-x_0\|<\delta
\Rightarrow
\|f(x)-L\|<\varepsilon.\]

\section{Sequential Characterization}\label{sequential-characterization}

\index{sequential characterization@Sequential Characterization}

\(x_0\) is an accumulation point of \(A\):

\[\lim_{x\to x_0}f(x)=L
\iff
f(x_k)\to L\] for every sequence \(x_k\in A\setminus\{x_0\}\) with
\(x_k\to x_0\).

\section{Uniqueness of the Limit}\label{uniqueness-of-the-limit}

\index{uniqueness of the limit@Uniqueness of the Limit}

If the limit exists, it is unique.

\section{Algebra of Limits}\label{algebra-of-limits}

\index{algebra of limits@Algebra of Limits}

\(f\to a\) and \(g\to b\):

\[f+g\to a+b,
\qquad
fg\to ab,\] \[\frac{f}{g}\to\frac ab
\quad\text{if }b\neq0\text{ and }g\neq0\text{ locally}.\] For vector
functions the limit is component by component.

\section{Iterated Limits and Joint
Limits}\label{iterated-limits-and-joint-limits}

\index{iterated limits and joint limits@Iterated Limits and Joint Limits}

In general:

\[\lim_{x\to x_0}\lim_{y\to y_0}f(x,y)\] and
\[\lim_{(x,y)\to(x_0,y_0)}f(x,y)\] are different concepts.

If the joint limit exists, every limit along an admissible curve or
sequence coincide with it. Limits along individual matching curves are
not sufficient, in general, to ensure the existence of the joint limit.

\bookmarksetup{startatroot}

\chapter{Continuity}\label{continuity}

\index{continuity@Continuity}

\(f\) is continuous \(x_0\in A\) if \[\lim_{x\to x_0}f(x)=f(x_0).\]
Equivalently: \[\forall\varepsilon>0\ \exists\delta>0:
\|x-x_0\|<\delta
\Rightarrow
\|f(x)-f(x_0)\|<\varepsilon.\]

\section{Sequential
Characterization}\label{sequential-characterization-1}

\index{sequential characterization@Sequential Characterization}

\[f\text{ continuous at }x_0
\iff
x_k\to x_0\Rightarrow f(x_k)\to f(x_0).\]

\section{Operations on Continuous
Functions}\label{operations-on-continuous-functions}

\index{operations on continuous functions@Operations on Continuous Functions}

Sum, product, composition and quotient with nonzero denominator of
continuous functions are continuous.

\section{Uniform Continuity}\label{uniform-continuity}

\index{uniform continuity@Uniform Continuity}

\(f:A\to\mathbb{R}^m\) is uniformly continuous if
\[\forall\varepsilon>0\ \exists\delta>0:\
\|x-y\|<\delta
\Rightarrow
\|f(x)-f(y)\|<\varepsilon\] for each \(x,y\in A\).

Conceptual difference:

\begin{itemize}
\item
  continuity: \(\delta\) can depend on the point;
\item
  uniform continuity: \(\delta\) depends only on \(\varepsilon\).
\end{itemize}

\section{Heine--Cantor Theorem}\label{heinecantor-theorem}

\index{heine-cantor theorem@Heine–Cantor Theorem}

\(K\subset\mathbb{R}^n\) is compact and \(f:K\to\mathbb{R}^m\) it is
continuous, then \(f\) is uniformly continuous.

\section{Weierstrass Theorem}\label{weierstrass-theorem}

\index{weierstrass theorem@Weierstrass Theorem}

\(K\subset\mathbb{R}^n\) is compact and \(f:K\to\mathbb{R}\) it is
continuous, then \(f\) is limited and attains its maximum and minimum.

\bookmarksetup{startatroot}

\chapter{Lipschitz and Hölder
Continuity}\label{lipschitz-and-huxf6lder-continuity}

\index{lipschitz and holder continuity@Lipschitz and Hölder Continuity}

\section{Lipschitz Continuity}\label{lipschitz-continuity}

\index{lipschitz continuity@Lipschitz Continuity}

\(f:A\to\mathbb{R}^m\) is Lipschitz if it exists \(L\ge0\) such that
\[\|f(x)-f(y)\|\le L\|x-y\|
\qquad\forall x,y\in A.\] You can define \[\operatorname{Lip}(f;A)=
\sup_{x\neq y}
\frac{\|f(x)-f(y)\|}{\|x-y\|}.\]

Each Lipschitz function is uniformly continuous.

\section{Local Lipschitz Continuity}\label{local-lipschitz-continuity}

\index{local lipschitz continuity@Local Lipschitz Continuity}

\(f\) is locally Lipschitz if every point has a neighborhood on which
\(f\) is Lipschitz.

\section{A Sufficient Derivative
Condition}\label{a-sufficient-derivative-condition}

\index{a sufficient derivative condition@A Sufficient Derivative Condition}

\(A\) It is convex, \(f\in C^1(A)\) and \[\sup_{x\in A}\|Df(x)\|\le L,\]
then \[\|f(x)-f(y)\|\le L\|x-y\|.\]

In a variable:

\[|f'(x)|\le L
\quad\Rightarrow\quad
|f(x)-f(y)|\le L|x-y|.\]

\section{Hölder Continuity}\label{huxf6lder-continuity}

\index{holder continuity@Hölder Continuity}

For \(0<\alpha\le1\), \(f\) is Hölder of exponent \(\alpha\) if
\[\|f(x)-f(y)\|
\le C\|x-y\|^\alpha.\] The case \(\alpha=1\) coincides with Lipschitz.

\bookmarksetup{startatroot}

\chapter{Derivatives in One Variable}\label{derivatives-in-one-variable}

\index{derivatives in one variable@Derivatives in One Variable}

For \(f:I\subset\mathbb{R}\to\mathbb{R}^m\):

\[f'(x_0)=
\lim_{h\to0}
\frac{f(x_0+h)-f(x_0)}{h}.\] Equivalently:

\[f(x_0+h)=f(x_0)+f'(x_0)h+o(h).\]

\section{Differentiability Implies
Continuity}\label{differentiability-implies-continuity}

\index{differentiability implies continuity@Differentiability Implies Continuity}

\[f\text{ differentiable at }x_0
\Longrightarrow
f\text{ continuous at }x_0.\] The converse is false.

\section{Differentiation Rules}\label{differentiation-rules}

\index{differentiation rules@Differentiation Rules}

\[(f+g)'=f'+g',\] \[(fg)'=f'g+fg',\] \[\left(\frac fg\right)'=
\frac{f'g-fg'}{g^2},\] \[(f\circ g)'(x)=f'(g(x))g'(x).\]

\section{Rolle's Theorem}\label{rolles-theorem}

\index{rolle� s theorem@Rolle’s Theorem}

\(f\in C([a,b])\), differentiable on \((a,b)\) and \(f(a)=f(b)\), then
there exists \(\xi\in(a,b)\) with \[f'(\xi)=0.\]

\section{Lagrange's Mean Value
Theorem}\label{lagranges-mean-value-theorem}

\index{lagrange� s mean value theorem@Lagrange’s Mean Value Theorem}

\(f\in C([a,b])\) and is differentiable on \((a,b)\), then there exists
\(\xi\in(a,b)\) such that \[f(b)-f(a)=f'(\xi)(b-a).\]

\section{Cauchy's Mean Value Theorem}\label{cauchys-mean-value-theorem}

\index{cauchy� s mean value theorem@Cauchy’s Mean Value Theorem}

Under the usual assumptions:

\[[f(b)-f(a)]g'(\xi)
=
[g(b)-g(a)]f'(\xi).\]

\bookmarksetup{startatroot}

\chapter{Partial and Directional
Derivatives}\label{partial-and-directional-derivatives}

\index{partial and directional derivatives@Partial and Directional Derivatives}

\section{Partial Derivative}\label{partial-derivative}

\index{partial derivative@Partial Derivative}

For \(f:\mathbb{R}^n\to\mathbb{R}^m\):

\[\frac{\partial f}{\partial x_i}(x)
=
\lim_{h\to0}
\frac{f(x+he_i)-f(x)}{h}.\]

\section{Directional Derivative}\label{directional-derivative}

\index{directional derivative@Directional Derivative}

In the direction \(v\):

\[D_vf(x)=
\lim_{t\to0}
\frac{f(x+tv)-f(x)}{t}.\]

For \(f:\mathbb{R}^n\to\mathbb{R}\) differentiable:
\[D_vf(x)=\nabla f(x)^Tv.\]

\(\|v\|=1\): \[|D_vf(x)|\le\|\nabla f(x)\|,\] with maximum in the
direction of gradient.

\section{Existence of Directional
Derivatives}\label{existence-of-directional-derivatives}

\index{existence of directional derivatives@Existence of Directional Derivatives}

The existence of all directional derivatives at one point does not
generally imply differentiability at the point.

\bookmarksetup{startatroot}

\chapter{Differentiability in Several
Variables}\label{differentiability-in-several-variables}

\index{differentiability in several variables@Differentiability in Several Variables}

\section{Fréchet Derivative}\label{fruxe9chet-derivative}

\index{frechet derivative@Fréchet Derivative}

\(f:\mathbb{R}^n\to\mathbb{R}^m\) is differentiable in \(x\) if there is
a linear application \[Df(x):\mathbb{R}^n\to\mathbb{R}^m\] such that
\[f(x+h)=f(x)+Df(x)h+o(\|h\|).\] Equivalently:

\[\lim_{h\to0}
\frac{\|f(x+h)-f(x)-Df(x)h\|}{\|h\|}=0.\]

\(Df(x)\) is unique.

\section{Jacobian}\label{jacobian}

\index{jacobian@Jacobian}

For \(f=(f_1,\dots,f_m)\):

\[Df(x)=J_f(x)=
\begin{pmatrix}
\partial_1f_1&\cdots&\partial_nf_1\\
\vdots&&\vdots\\
\partial_1f_m&\cdots&\partial_nf_m
\end{pmatrix}.\]

For \(m=1\):

\[Df(x)=\nabla f(x)^T.\]

\section{Differentiability and Partial
Derivatives}\label{differentiability-and-partial-derivatives}

\index{differentiability and partial derivatives@Differentiability and Partial Derivatives}

\(f\) is differentiable, then all partial derivatives exist.

The converse is false in general.

Classic enough condition:
\[\partial_i f_j\text{ exist in a neighborhood and are continuous at }x_0
\Longrightarrow
f\text{ differentiable at }x_0.\] In particular:

\[f\in C^1
\Longrightarrow
f\text{ differentiable}.\]

\section{Differentiability Implies
Continuity}\label{differentiability-implies-continuity-1}

\index{differentiability implies continuity@Differentiability Implies Continuity}

\[f\text{ differentiable at }x_0
\Longrightarrow
f\text{ continuous at }x_0.\]

\bookmarksetup{startatroot}

\chapter{Gâteaux versus Fréchet
Differentiability}\label{guxe2teaux-versus-fruxe9chet-differentiability}

\index{gateaux versus frechet differentiability@Gâteaux versus Fréchet Differentiability}

\section{Gâteaux Derivative}\label{guxe2teaux-derivative}

\index{gateaux derivative@Gâteaux Derivative}

The derivative of Gâteaux \(f\) in \(x\) in the direction \(h\) It is
\[D_Gf(x)[h]
=
\lim_{t\to0}
\frac{f(x+th)-f(x)}{t}.\]

If it exists for each \(h\) and is linear in \(h\), we talk about
Gâteaux\textquotesingle s differentiability.

\section{Relationships}\label{relationships}

\index{relationships@Relationships}

\[\text{Fréchet differentiable}
\Longrightarrow
\text{Gâteaux differentiable}.\] In finite dimensions, under appropriate
assumptions of continuity of the derivative, the differentiability of
Gâteaux can imply that of Fréchet.

The only existence of directional derivatives does not guarantee neither
linearity nor differentiability of Fréchet.

\bookmarksetup{startatroot}

\chapter{Chain Rule and Derivatives of
Compositions}\label{chain-rule-and-derivatives-of-compositions}

\index{chain rule and derivatives of compositions@Chain Rule and Derivatives of Compositions}

\[f:\mathbb{R}^n\to\mathbb{R}^m,
\qquad
g:\mathbb{R}^m\to\mathbb{R}^p\] are differentiable:
\[D(g\circ f)(x)=Dg(f(x))Df(x).\]

For \(g:\mathbb{R}^m\to\mathbb{R}\):

\[\nabla(g\circ f)(x)
=
Df(x)^T\nabla g(f(x)).\]

\(x=x(t)\):

\[\frac{d}{dt}f(x(t))
=
Df(x(t))x'(t).\]

For \(f:\mathbb{R}^n\to\mathbb{R}\):

\[\frac{d}{dt}f(x(t))
=
\nabla f(x(t))^Tx'(t).\]

\bookmarksetup{startatroot}

\chapter{Second Derivatives and the
Hessian}\label{second-derivatives-and-the-hessian}

\index{second derivatives and the hessian@Second Derivatives and the Hessian}

\section{Hessian}\label{hessian}

\index{hessian@Hessian}

For \(f:\mathbb{R}^n\to\mathbb{R}\) twice differentiable:
\[\operatorname{Hess}f(x)=\nabla^2f(x)=
\left[
\frac{\partial^2f}{\partial x_i\partial x_j}
\right]_{i,j}.\]

\section{Schwarz--Clairaut Theorem}\label{schwarzclairaut-theorem}

\index{schwarz-clairaut theorem@Schwarz–Clairaut Theorem}

If the second mixed derivatives are continuous in a neighborhood:
\[\frac{\partial^2f}{\partial x_i\partial x_j}
=
\frac{\partial^2f}{\partial x_j\partial x_i}.\] So Hessian is symmetric.

\section{Second Directional
Derivative}\label{second-directional-derivative}

\index{second directional derivative@Second Directional Derivative}

\(f\in C^2\):

\[D_v^2f(x)=v^T\operatorname{Hess}f(x)v.\]

\section{Second Derivative of a Composition along a
Curve}\label{second-derivative-of-a-composition-along-a-curve}

\index{second derivative of a composition along a curve@Second Derivative of a Composition along a Curve}

\(x=x(t)\) and \(f:\mathbb{R}^n\to\mathbb{R}\):

\[\frac{d^2}{dt^2}f(x(t))
=
x'(t)^T\operatorname{Hess}f(x(t))x'(t)
+\nabla f(x(t))^Tx''(t).\]

\bookmarksetup{startatroot}

\chapter{Regularity Classes}\label{regularity-classes}

\index{regularity classes@Regularity Classes}

\[C^0=\text{continuous functions},\]
\[C^1=\text{functions with continuous first derivatives},\]
\[C^k=\text{functions with continuous derivatives up to order }k\text{ continuous},\]
\[C^\infty=\bigcap_{k\ge0}C^k.\]

An analytical function locally admits a development in series of powers
converging to function.

In general:

\[\text{analytic}\Longrightarrow C^\infty,\] but vice versa is false.

\bookmarksetup{startatroot}

\chapter{\texorpdfstring{Asymptotic Notation: Little \(o\) and Big
\(O\)}{Asymptotic Notation: Little o and Big O}}\label{asymptotic-notation-little-o-and-big-o}

\index{asymptotic notation little o and big o@Asymptotic Notation: Little o and Big O}

\section{\texorpdfstring{Little \(o\)}{Little o}}\label{little-o}

\index{little o@Little o}

For \(x\to x_0\):

\[f(x)=o(g(x))\] if \[\frac{f(x)}{g(x)}\to0\] when the report is defined
locally.

Equivalently:

\[f=o(g)
\iff
\forall\varepsilon>0\ \exists\delta>0:
|f(x)|\le\varepsilon|g(x)|\] for \(x\) sufficiently close to \(x_0\).

\section{\texorpdfstring{Big \(O\)}{Big O}}\label{big-o}

\index{big o@Big O}

\[f(x)=O(g(x))\] if they exist \(C>0\) and a neighborhood of \(x_0\)
such as \[|f(x)|\le C|g(x)|.\]

\section{\texorpdfstring{Relationship between \(o\) and
\(O\)}{Relationship between o and O}}\label{relationship-between-o-and-o}

\index{relationship between o and o@Relationship between o and O}

\[f=o(g)\Longrightarrow f=O(g).\] The converse is false.

\section{Asymptotic Equivalence}\label{asymptotic-equivalence}

\index{asymptotic equivalence@Asymptotic Equivalence}

\[f(x)\sim g(x)
\quad\Longleftrightarrow\quad
\frac{f(x)}{g(x)}\to1.\] Equivalently:

\[f=g+o(g).\]

\section{\texorpdfstring{\(\Theta\)
Notation}{\textbackslash Theta Notation}}\label{theta-notation}

\index{theta notation@Theta Notation}

\[f=\Theta(g)\] if \[c|g(x)|\le|f(x)|\le C|g(x)|\] locally, for some
\(c,C>0\).

\section{Algebraic Rules}\label{algebraic-rules}

\index{algebraic rules@Algebraic Rules}

\(f=o(g)\) and \(h=O(k)\):

\[fh=o(gk).\]

\(f_1=o(g)\)and\(f_2=o(g)\): \[f_1+f_2=o(g).\]

\(f=O(g)\)and\(h=O(k)\):

\[fh=O(gk).\]

\(f=o(1)\): \[1+f=1+o(1).\]

\section{Vector-Valued Version}\label{vector-valued-version}

\index{vector-valued version@Vector-Valued Version}

For \(f:\mathbb{R}^n\to\mathbb{R}^m\):

\[f(h)=o(\|h\|^k)\] means \[\frac{\|f(h)\|}{\|h\|^k}\to0.\] Similarly:

\[f(h)=O(\|h\|^k)
\iff
\|f(h)\|\le C\|h\|^k.\]

\bookmarksetup{startatroot}

\chapter{Elementary Asymptotic
Expansions}\label{elementary-asymptotic-expansions}

\index{elementary asymptotic expansions@Elementary Asymptotic Expansions}

For \(x\to0\):

\[e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+O(x^4),\]
\[\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+O(x^4),\]
\[\sin x=x-\frac{x^3}{6}+O(x^5),\]
\[\cos x=1-\frac{x^2}{2}+\frac{x^4}{24}+O(x^6),\] \[(1+x)^\alpha
=1+\alpha x+\frac{\alpha(\alpha-1)}{2}x^2+O(x^3).\] In particular:
\[\sin x\sim x,
\qquad
\tan x\sim x,
\qquad
1-\cos x\sim\frac{x^2}{2},\] \[e^x-1\sim x,
\qquad
\log(1+x)\sim x.\]

\bookmarksetup{startatroot}

\chapter{Taylor's Theorem in One
Variable}\label{taylors-theorem-in-one-variable}

\index{taylor� s theorem in one variable@Taylor’s Theorem in One Variable}

Both \(f\) sufficiently regular near to \(x_0\).
Taylor\textquotesingle s polynomial of order \(p\) It is \[T_p(x;x_0)
=
\sum_{k=0}^p
\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k.\]

\section{Peano Remainder}\label{peano-remainder}

\index{peano remainder@Peano Remainder}

\(f\) It is \(p\) times differentiable in an appropriate sense:

\[f(x)
=
\sum_{k=0}^p
\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k
+o(|x-x_0|^p).\]

\section{Lagrange Remainder}\label{lagrange-remainder}

\index{lagrange remainder@Lagrange Remainder}

\(f\in C^{p+1}\) between \(x_0\) and \(x\):

\[f(x)=T_p(x;x_0)
+
\frac{f^{(p+1)}(\xi)}{(p+1)!}(x-x_0)^{p+1}\] for some \(\xi\) between
\(x_0\) and \(x\).

\section{Integral Remainder}\label{integral-remainder}

\index{integral remainder@Integral Remainder}

\(f^{(p+1)}\) is continuous:

\[f(x)=T_p(x;x_0)
+
\frac1{p!}
\int_{x_0}^x
f^{(p+1)}(t)(x-t)^p\,dt.\]

\section{\texorpdfstring{Big-\(O\) Form}{Big-O Form}}\label{big-o-form}

\index{big-o form@Big-O Form}

\(f^{(p+1)}\) is locally limited:

\[f(x)=T_p(x;x_0)+O(|x-x_0|^{p+1}).\]

\bookmarksetup{startatroot}

\chapter{Taylor's Theorem in Several
Variables}\label{taylors-theorem-in-several-variables}

\index{taylor� s theorem in several variables@Taylor’s Theorem in Several Variables}

Both \(f:\mathbb{R}^n\to\mathbb{R}\). With \(h=x-x_0\):

\section{First Order}\label{first-order}

\index{first order@First Order}

\(f\) is differentiable:

\[f(x_0+h)
=
f(x_0)+\nabla f(x_0)^Th+o(\|h\|).\]

\(Df\) is locally Lipschitz:

\[f(x_0+h)
=
f(x_0)+\nabla f(x_0)^Th+O(\|h\|^2).\]

\section{Second Order}\label{second-order}

\index{second order@Second Order}

\(f\) is twice differentiable with sufficient regularity:

\[f(x_0+h)
=
f(x_0)
+\nabla f(x_0)^Th
+\frac12h^T\operatorname{Hess}f(x_0)h
+o(\|h\|^2).\]

\(\operatorname{Hess}f\) is locally Lipschitz:

\[f(x_0+h)
=
f(x_0)
+\nabla f(x_0)^Th
+\frac12h^T\operatorname{Hess}f(x_0)h
+O(\|h\|^3).\]

\section{Form along a Line Segment}\label{form-along-a-line-segment}

\index{form along a line segment@Form along a Line Segment}

Defining \[\phi(t)=f(x_0+th),\] Taylor multidimensional is reduced to
Taylor 1D \(\phi\). In particular:

\[\phi'(t)=\nabla f(x_0+th)^Th,\]
\[\phi''(t)=h^T\operatorname{Hess}f(x_0+th)h.\]

\section{Second-Order Lagrange
Remainder}\label{second-order-lagrange-remainder}

\index{second-order lagrange remainder@Second-Order Lagrange Remainder}

\(f\in C^2\) the segment between \(x_0\) and \(x_0+h\), exist
\(\theta\in(0,1)\) that the \[f(x_0+h)
=
f(x_0)+\nabla f(x_0)^Th
+\frac12h^T\operatorname{Hess}f(x_0+\theta h)h.\]

\section{Multi-Index Form}\label{multi-index-form}

\index{multi-index form@Multi-Index Form}

For \(\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb{N}^n\):
\[|\alpha|=\sum_i\alpha_i,
\qquad
\alpha!=\prod_i\alpha_i!,\] \[h^\alpha=\prod_i h_i^{\alpha_i},
\qquad
D^\alpha f
=
\frac{\partial^{|\alpha|}f}
{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}.\] Then formally:
\[f(x_0+h)
=
\sum_{|\alpha|\le p}
\frac{D^\alpha f(x_0)}{\alpha!}h^\alpha
+R_p(h).\] With appropriate hypotheses:

\[R_p(h)=o(\|h\|^p),\] or, with a more locally limited derivative,
\[R_p(h)=O(\|h\|^{p+1}).\]

\bookmarksetup{startatroot}

\chapter{Mean Value Theorem in Several
Variables}\label{mean-value-theorem-in-several-variables}

\index{mean value theorem in several variables@Mean Value Theorem in Several Variables}

For \(f:\mathbb{R}^n\to\mathbb{R}\), if the segment
\[[x,y]=\{x+t(y-x):0\le t\le1\}\] is contained in the domain and \(f\)
is differentiable, exists \(\xi\) on the segment
\[f(y)-f(x)=\nabla f(\xi)^T(y-x).\]

For \(f:\mathbb{R}^n\to\mathbb{R}^m\) there is generally no analogue
with a single point \(\xi\) valid simultaneously for all components. The
estimate is:

\[\|f(y)-f(x)\|
\le
\sup_{z\in[x,y]}\|Df(z)\|\,\|y-x\|.\]

\bookmarksetup{startatroot}

\chapter{Functions with Lipschitz
Gradient}\label{functions-with-lipschitz-gradient}

\index{functions with lipschitz gradient@Functions with Lipschitz Gradient}

\(f:\mathbb{R}^n\to\mathbb{R}\) It is \(L\)-smooth se
\[\|\nabla f(x)-\nabla f(y)\|
\le L\|x-y\|.\]

\(f\in C^2\) on a convex set:
\[\|\operatorname{Hess}f(x)\|\le L\quad\forall x\] implies \(\nabla f\)
It is \(L\)-Lipschitz.

\section{Descent Lemma}\label{descent-lemma}

\index{descent lemma@Descent Lemma}

\(\nabla f\) It is \(L\)-Lipschitz:

\[f(y)
\le
f(x)+\nabla f(x)^T(y-x)
+\frac L2\|y-x\|^2.\] More precisely:

\[|f(y)-f(x)-\nabla f(x)^T(y-x)|
\le
\frac L2\|y-x\|^2.\]

\bookmarksetup{startatroot}

\chapter{Inverse Function Theorem}\label{inverse-function-theorem}

\index{inverse function theorem@Inverse Function Theorem}

Both \(f:\mathbb{R}^n\to\mathbb{R}^n\) class \(C^1\) in a neighborhood
of \(x_0\). \[\det Df(x_0)\neq0,\] then \(f\) is locally invertible near
\(x_0\) and the inverse is \(C^1\). In addition:

\[D(f^{-1})(f(x_0))
=
Df(x_0)^{-1}.\]

\bookmarksetup{startatroot}

\chapter{Implicit Function Theorem}\label{implicit-function-theorem}

\index{implicit function theorem@Implicit Function Theorem}

Both \[F:\mathbb{R}^n\times\mathbb{R}^m\to\mathbb{R}^m\] class \(C^1\),
with \[F(x_0,y_0)=0.\] \[D_yF(x_0,y_0)\] is invertible, then locally
there is a function \[y=\varphi(x)\] such that \[F(x,\varphi(x))=0.\]
The derivative is \[D\varphi(x)
=
-[D_yF(x,\varphi(x))]^{-1}D_xF(x,\varphi(x)).\]

\bookmarksetup{startatroot}

\chapter{Unconstrained Extrema and the Second-Order
Test}\label{unconstrained-extrema-and-the-second-order-test}

\index{unconstrained extrema and the second-order test@Unconstrained Extrema and the Second-Order Test}

\(f:\mathbb{R}^n\to\mathbb{R}\) is differentiable and \(x_*\) is an
extreme internal place:

\[\nabla f(x_*)=0.\]

\(f\in C^2\) and \(\nabla f(x_*)=0\):

\begin{itemize}
\item
  \(\operatorname{Hess}f(x_*)\succ0\) \(\Rightarrow\) minimum local
  narrow;
\item
  \(\operatorname{Hess}f(x_*)\prec0\) \(\Rightarrow\) maximum local
  tight;
\item
  Hessian indefinite \(\Rightarrow\) saddle point;
\item
  Hessian semidefinite \(\Rightarrow\) inconclusive tests in general.
\end{itemize}

\bookmarksetup{startatroot}

\chapter{Table of Main Implications}\label{table-of-main-implications}

\index{table of main implications@Table of Main Implications}

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\linewidth - 2\tabcolsep) * \real{0.5000}}
  >{\raggedright\arraybackslash}p{(\linewidth - 2\tabcolsep) * \real{0.5000}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
\textbf{Assumption}
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
\textbf{Consequence}
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Fréchet differentiable in \(x_0\) & continues \(x_0\) \\
\(C^1\) close to \(x_0\) & differentiable in \(x_0\) \\
Lipschitz & uniformly continuous \\
\(Df\) limited on convex set & \(f\) Lipschitz \\
\(\nabla f\) \(L\)-Lipschitz & linear rest \(O(\Vert h\Vert ^2)\) \\
\(\operatorname{Hess}f\) Lipschitz & Taylor order 2 with rest
\(O(\Vert h\Vert ^3)\) \\
\(f=o(g)\) & \(f=O(g)\) \\
\(f\sim g\) & \(f=g+o(g)\) \\
\(f\in C^2\) & Hessian symmetric \\
\(\det Df(x_0)\neq0\) & local invertability \\
\end{longtable}
}

\bookmarksetup{startatroot}

\chapter{Hierarchy of Regularity
Notions}\label{hierarchy-of-regularity-notions}

\index{hierarchy of regularity notions@Hierarchy of Regularity Notions}

In general:

\[C^1
\Longrightarrow
\text{differentiable}
\Longrightarrow
\text{continuous}.\]

On convex sets, if \(Df\) is limited:

\[C^1+\sup\|Df\|<\infty
\Longrightarrow
\text{Lipschitz}.\]

For \(0<\alpha<1\):

\[\text{Lipschitz}
\Longrightarrow
C^{0,\alpha}\text{ locally on bounded sets}.\]

But, in general, none of the inverse implications is valid without
additional hypotheses.

\bookmarksetup{startatroot}

\chapter{Quick Reference for Taylor Expansions and
Orders}\label{quick-reference-for-taylor-expansions-and-orders}

\index{quick reference for taylor expansions and orders@Quick Reference for Taylor Expansions and Orders}

{\def\LTcaptype{none} % do not increment counter
\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\linewidth - 2\tabcolsep) * \real{0.5000}}
  >{\raggedright\arraybackslash}p{(\linewidth - 2\tabcolsep) * \real{0.5000}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
\textbf{Regularity}
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
\textbf{Local expansion}
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Differentiated & \(f(x+h)=f(x)+Df(x)h+o(\Vert h\Vert )\) \\
\(Df\) locally Lipschitz & \(f(x+h)=f(x)+Df(x)h+O(\Vert h\Vert ^2)\) \\
Twice differentiable &
\(f(x+h)=f(x)+Df(x)h+\frac12D^2f(x)[h,h]+o(\Vert h\Vert ^2)\) \\
\(D^2f\) locally Lipschitz &
\(f(x+h)=f(x)+Df(x)h+\frac12D^2f(x)[h,h]+O(\Vert h\Vert ^3)\) \\
\(C^{p+1}\) locally & \(f=T_p+O(\Vert h\Vert ^{p+1})\) \\
Taylor-Peano order \(p\) & \(f=T_p+o(\Vert h\Vert ^p)\) \\
\end{longtable}
}

%----------------------------------------------------------------------------------------
% ANALYTICAL INDEX
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\end{document}