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\newtheorem*{objectives}{Objectives}
\newtheorem*{requirements}{Requirements}

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\begin{document}

\mytitle[Modulus of continuity with respect to the logarithmic distance]%
{Modulus of continuity of functions\\with respect to the logarithmic distance}

\medskip

\begin{objectives}
Study some properties of the \emph{modulus of continuity} $\Omega_{\rho,f}$,
\[
\Omega_{\rho,f}(\delta)
\eqdef\sup\bigl\{|f(x)-f(y)|\colon\quad x,y\in\PositiveRealNumbers,\quad\rho(x,y)\le\delta\bigr\},
\]
of a function $f\colon\PositiveRealNumbers\to\ComplexNumbers$
with respect to the \emph{logarithmic distance}
$\rho\colon\PositiveRealNumbers\times\PositiveRealNumbers\to[0,+\infty)$ defined by
\[
\rho(x,y)\eqdef\bigl|\ln(x)-\ln(y)\bigr|.
\]
\end{objectives}

\begin{requirements}
Properties of the logarithmic distance $\rho$.
\end{requirements}

\bigskip

\section*{Logarithmic distance (review)}

\medskip

\begin{exercise}
Fill the table:
\par\smallskip
\[
\begin{array}{c|c|c|c}
\vertstrut & \hsp{1em}\max(x,y)\hsp{1em} & \hsp{1em}\min(x,y)\hsp{1em} & 
  \hsp{3em}\displaystyle\bigl|\ln(x)-\ln(y)\bigr|\hsp{3em} \\\hline
\vertstrut \text{Case}\ x\ge y\colon & & & \\\hline
\vertstrut \text{Case}\ x<y\colon & & & 
\end{array}
\]
\end{exercise}

\bigskip

\begin{exercise}
Express $\rho(x,y)$ through $\max(x,y)$ and $\min(x,y)$:
\par\smallskip
\[
\rho(x,y)
\;=\;\ln\bigl(\hsp{6em}\bigr)-\ln\bigl(\hsp{6em}\bigr)
\;=\;\ln\frac{\hsp{7em}}{\hsp{7em}}.
\]
\end{exercise}

\bigskip

\begin{exercise}[$\rho$ is invariant with respect to the dilations]
Let $x,y,t\in\PositiveRealNumbers$. Simplify:
\par\smallskip
\[
\rho(tx,ty)=\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise} \label{exer:rhoxy_through_rhoxyone}
Let $x,y\in\PositiveRealNumbers$. Then
\par\smallskip
\[
\rho(x,y)=\rho\bigl(\hsp{4em},\ 1\bigr).
\]
\end{exercise}


\clearpage
\section*{$\boldsymbol{\delta}$-neighborhoods of $\boldsymbol{1}$ with respect to $\boldsymbol{\rho}$ (review)}

\medskip

\begin{exercise}[left $\delta$-neighborhood of $1$] \mbox{}\\[1ex]
Let $\delta>0$.
Find all $u\in(0,1]$ such that $\rho(u,1)\le\delta$.
\end{exercise}

\bigskip\bigskip\bigskip

\begin{exercise}[right $\delta$-neighborhood of $1$] \mbox{}\\[1ex]
Let $\delta>0$.
Find all $u\in[1,+\infty)$ such that $\rho(u,1)\le\delta$.
\end{exercise}

\bigskip\bigskip\bigskip

\begin{exercise}[$\delta$-neighborhood of $1$] \mbox{}\\[1ex] \label{exer:delta_neighborhood}
Let $\delta>0$.
Find all $u>0$ such that $\rho(u,1)\le\delta$.
\end{exercise}

\bigskip\bigskip\bigskip


\bigskip

\section*{$\boldsymbol{\delta}$-entourages with respect to the distance $\boldsymbol{\rho}$ (review)}

\medskip

\begin{exercise} \label{exer:right_delta_entourages}
Let $\delta>0$ and $x\in\PositiveRealNumbers$.
Find all $y\in[x,+\infty)$ such that $\rho(x,y)\le\delta$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise} \label{exer:left_delta_entourages}
Let $\delta>0$ and $x\in\PositiveRealNumbers$.
Find all $y\in(0,x]$ such that $\rho(x,y)\le\delta$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise} \label{exer:delta_entourages}
Let $\delta>0$ and $x\in\PositiveRealNumbers$.
Find all $y\in\PositiveRealNumbers$ such that $\rho(x,y)\le\delta$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip


\clearpage
\section*{Definition of the modulus of continuity of a function\\
with respect to the logarithmic distance}

\medskip

\begin{defn}[modulus of continuity of a function with respect to the logarithmic distance]
Let $f\colon\PositiveRealNumbers\to\ComplexNumbers$.
Define $\Omega_{\rho,f}\colon\PositiveRealNumbers\to[0,+\infty]$ by
\[
\Omega_{\rho,f}(\delta)\eqdef\sup\bigl\{|f(x)-f(y)|\colon\quad x,y>0,\quad\rho(x,y)\le\delta\bigr\}.
\]
\end{defn}

\bigskip

\begin{exercise}
Write another description of $\Omega_{\rho,f}$
using the result of the Exercise \ref{exer:rhoxy_through_rhoxyone}:
\par\smallskip
\begin{equation} \label{eq:Omega_from_rho_to_one}
\Omega_{\rho,f}(\delta)
=\sup\Bigl\{|f(x)-f(y)|\colon\quad x,y>0,\quad \rho\Bigl(\ubrace{3em},1\Bigr)\le\delta\Bigr\}.
\end{equation}
\end{exercise}

\bigskip

\begin{exercise}
Rewrite \eqref{eq:Omega_from_rho_to_one}
using the result of the Exercise \ref{exer:delta_neighborhood}:
\par\smallskip
\[
\Omega_{\rho,f}(\delta)
=\sup\bigl\{|f(x)-f(y)|\colon\quad x,y>0,\quad \ubrace{3em}\le\ubrace{3em}\le\ubrace{3em}\bigr\}.
\]
\end{exercise}

\bigskip

\begin{exercise}
Apply the description of the $\delta$-entourages
found in the Exercise \ref{exer:delta_entourages}:
\par\smallskip
\[
\Omega_{\rho,f}(\delta)
=\sup\bigl\{|f(x)-f(y)|\colon\quad x>0,\quad\ubrace{3em}\le y\le\ubrace{3em}\bigr\}.
\]
\end{exercise}

\bigskip

\begin{exercise}
In other words, $\Omega_{\rho,f}$ can be written as a double supremum:
\par\smallskip
\[
\Omega_{\rho,f}(\delta)=\sup_{x>0}\,\sup_{\hsp{3em}\le y\le\hsp{3em}}|f(x)-f(y)|.
\]
\end{exercise}

\bigskip


\clearpage
\section*{Applying the symmetricity\\
in the definition of the modulus of continuity}

\medskip\noindent
In the exercises of this section we fix a function
$f\colon\PositiveRealNumbers\to\ComplexNumbers$
and a number $\delta\in(0,1)$.

\noindent
Consider the sets
\begin{align*}
S&\eqdef\bigl\{|f(x)-f(y)|\colon\quad x,y>0,\quad \rho(x,y)\le\delta\bigr\},
\\[1ex]
S_1&\eqdef\bigl\{|f(a)-f(b)|\colon\quad a,b>0,\quad a\le b,\quad\rho(a,b)\le\delta\bigr\},
\\[1ex]
S_2&\eqdef\bigl\{|f(u)-f(v)|\colon\quad u,v>0,\quad u\ge v,\quad\rho(u,v)\le\delta\bigr\}.
\end{align*}
We are going to find relations between the sets $S_1$, $S_2$ and $S$.

\bigskip

\begin{exercise}[recall the definition of total order] \mbox{}\\
A binary relation $\prec$ on a set $X$ is called a \emph{total order} (or \emph{linear order}) on $X$\\
if it has the following properties:
\begin{enumerate}
\item Transitivity:
\par\smallskip
\[
\forall x,y\in X\qquad\bigl((x \prec y\quad \wedge\quad y \prec x)\quad\Rightarrow\quad \ubrace{4em}\;\bigr).
\]
\item Antisymmetry:
\par\smallskip
\[
\forall x,y\in X\qquad\bigl((x \prec y\quad\wedge\quad y \prec x)\quad\Rightarrow\quad \ubrace{4em}\;\bigr).
\]
\item Totality:
\par\smallskip
\[
\forall x,y\in X\qquad\bigl(x\prec y\quad\ubrace{3em}\quad y\prec x\bigr).
\]
\end{enumerate}
\end{exercise}

\bigskip

\begin{exercise} \label{exer:S_through_S1_and_S2}
The usual non-string order $\le$ is a total order (= linear order) on $\PositiveRealNumbers$.\\[1ex]
It means that for all $x,y>0$ we have
\[
x\le y\qquad\ubrace{2em}\qquad x\ge y.
\]
Therefore $S$ can be expressed through $S_1$ and $S_2$ in the following manner:
\par\smallskip
\[
S = S_1 \ubrace{2em} S_2.
\]
\end{exercise}

\clearpage
%\begin{exercise}
%Copy the definitions of the sets $S$, $S_1$ and $S_2$:
%\begin{align*}
%S&\eqdef\bigl\{|f(x)-f(y)|\colon \hsp{15em}\bigr\},
%\\[2ex]
%S_1&\eqdef\bigl\{|f(a)-f(b)|\colon \hsp{20em}\bigr\},
%\\[2ex]
%S_2&\eqdef\bigl\{|f(u)-f(v)|\colon \hsp{20em}\bigr\}.
%\end{align*}
%\end{exercise}

\begin{exercise}
The expression $|f(x)-f(y)|$ is symmetric with respect to $x$ and $y$:
\par\smallskip
\[
|f(x)-f(y)|=\ubrace{7em}.
\]
The function $\rho$ is also symmetric: \quad $\rho(x,y)=\ubrace{5em}$.
\end{exercise}

\bigskip

\begin{exercise} \label{exer:relation_between_S1_and_S2}
Let $s\in S_1$. By the definition of $S_1$ there exist $a,b>0$ such that
\par\smallskip
\[
s=|f(a)-f(b)|,\qquad a\,\ubrace{2em}\,b,\qquad \rho(a,b)\ubrace{3em}.
\]
Apply the symmetricity of $|f(a)-f(b)|$:
\par\smallskip
\[
s=|f(a)-f(b)|=|f(\,\ubrace{2em}\,)-f(\,\ubrace{2em}\,)|,
\]
and the symmetricity of $\rho$:
\par\smallskip
\[
\rho(\,\ubrace{2em}\,,\,\ubrace{2em}\,)=\rho(a,b)\le\,\ubrace{2em}.
\]
Put\quad $u\eqdef\,\ubrace{2em}$,\quad $v\eqdef\,\ubrace{2em}$.\quad
Then by the previous formulas,
\par\smallskip
\[
s=\,\ubrace{6em}\qquad\text{and}\qquad \rho(\,\ubrace{2em}\,,\,\ubrace{2em}\,)\le\,\ubrace{2em}.
\]
By the definition of $S_2$ it means that \quad $\ubrace{4em}$.
\par\medskip\noindent
Since $s$ is an arbitrary element of $S_1$, the following inclusion holds:\quad $S_1 \ubrace{2em} S_2$.
\par\medskip\noindent
It can be proved in a similar manner that
\quad $S_2\,\ubrace{2em}\,S_1$.
\par\medskip\noindent
Conclusion:\quad $S_1\,\ubrace{2em}\,S_2$.
\end{exercise}

\bigskip

\begin{exercise} \label{exer:S_S1_S2}
Using the results of the Exercises \ref{exer:S_through_S1_and_S2}
and \ref{exer:relation_between_S1_and_S2}
establish a relation between the sets $S$, $S_1$, $S_2$.
\end{exercise}

\bigskip\bigskip\bigskip

\clearpage
\begin{exercise}
Express $\Omega_{\rho,f}(\delta)$ in terms of the set $S$:
\par\smallskip
\[
\Omega_{\rho,f}(\delta)=\ubrace{6em}.
\]
Taking in account the result of the Exercise \ref{exer:S_S1_S2}
we can express the modulus of continuity $\Omega_{\rho,f}$ through $S_1$ or through $S_2$:
\par\smallskip
\begin{equation} \label{eq:Omega_through_S1_and_through_S2}
\Omega_{\rho,f}(\delta)=\ubrace{6em}=\ubrace{6em}.
\end{equation}
\end{exercise}

\bigskip

\begin{exercise}
Substituting the definitions of $S_1$ and $S_2$ into \eqref{eq:Omega_through_S1_and_through_S2}
we obtain the following representations of the modulus of continuity $\Omega_{\rho,f}(\delta)$:
\par\medskip
\begin{align}
\Omega_{\rho,f}(\delta)
\,&=\,\sup\,\bigl\{\hsp{7em}\colon\quad
x,y\in\ubrace{2em},\qquad x\le y,\qquad \rho(x,y)\ubrace{3em}\bigr\}
\label{eq:Omega_f_x_le_y}
\\[3ex]
\Omega_{\rho,f}(\delta)
\,&=\,\sup\,\bigl\{\hsp{7em}\colon\quad
x,y\in\ubrace{2em},\qquad\phantom{ x\le y,\qquad \rho(x,y)\ubrace{3em}}\bigr\}.
\label{eq:Omega_f_x_ge_y}
\end{align}
\end{exercise}

\bigskip

\begin{exercise}
Using the results of the Exercises \ref{exer:right_delta_entourages} and \ref{exer:left_delta_entourages}
write the formulas \eqref{eq:Omega_f_x_le_y} and \eqref{eq:Omega_f_x_le_y} in the following manner:
\par\medskip
\begin{align}
\Omega_{\rho,f}(\delta)
\,&=\,\sup\,\bigl\{\hsp{7em}\colon\quad x>0,\qquad \ubrace{3em}\le y\le\ubrace{3em}\bigr\},
\label{eq:Omega_f_x_y}
\\[3ex]
\Omega_{\rho,f}(\delta)
\,&=\,\sup\,\bigl\{\hsp{7em}\colon\quad \phantom{x>0,\qquad \ubrace{3em}\le y\le\ubrace{3em}}\bigr\}.
\label{eq:Omega_f_y_x}
\end{align}
\end{exercise}

\bigskip

\begin{exercise}
Rewrite the formulas \eqref{eq:Omega_f_x_y} and \eqref{eq:Omega_f_y_x} using double supremums:
\par\medskip
\begin{align}
\Omega_{\rho,f}(\delta)
\,&=\,\sup_{x>0}\sup_{\hsp{3em}\le y\le\hsp{3em}}\,|f(x)-f(y)|,
\label{eq:Omega_f_sup_sup_x_le_y}
\\[3ex]
\Omega_{\rho,f}(\delta)\,&=\,
\label{eq:Omega_f_sup_sup_y_le_x}
\end{align}
\end{exercise}

\bigskip
\section*{The logarithmic distance and the standard distance}

\medskip

\begin{defn}[standard distance on the real line]
The standard distance $d\colon\RealNumbers\times\RealNumbers\to\ComplexNumbers$ is defined by:
\par\smallskip
\[
d(t,u)\,\eqdef\,\ubrace{4em}.
\]
\end{defn}

\bigskip

\begin{exercise}
Recall the definition of the logarithmic distance $\rho$:
\par\smallskip
\[
\rho(x,y)\,\eqdef\,\ubrace{7em}.
\]
\end{exercise}

\bigskip

\begin{exercise} \label{exer:logarithmic_change_of_variables_in_the_distance_rho}
The logarithmic distance $\rho$ on the positive half-line
can be expressed through the standard distance $d$ on the real line
in the following manner:
\par\smallskip
\[
\rho(x,y)\eqdef\ubrace{7em}=|t-u|,
\]
there
\par\smallskip
\[
t=\ubrace{4em},\qquad u=\ubrace{4em}.
\]
The variables $x$ and $y$ can be expressed through $t$ and $u$:
\par\smallskip
\begin{equation} \label{eq:x_and_y_through_t_and_u}
x=\ubrace{4em},\qquad y=\ubrace{4em}.
\end{equation}
\par\smallskip\noindent
The function $\ubrace{3em}$ is a bijection of $\PositiveRealNumbers$ onto $\ubrace{3em}$.
\par\bigskip\noindent
Therefore the domain of the variables $t$ and $u$ is the set $\ubrace{3em}$.
\end{exercise}

\clearpage
\section*{Expression through the standard modulus of continuity}

\medskip\noindent
We suppose that $f\colon\PositiveRealNumbers\to\ComplexNumbers$ and $\delta>0$.

\bigskip

\begin{exercise}
Copy the definition of $\Omega_{\rho,f}(\delta)$:
\par\smallskip
\begin{equation} \label{eq:Omega_rho_f_delta_definition_copy}
\Omega_{\rho,f}(\delta)\,\eqdef\,\sup\,\Bigl\{|f(x)-f(y)|\colon\quad \hsp{15em}\Bigr\}.
\end{equation}
\end{exercise}

\medskip

\begin{exercise}
In the right-hand side of \eqref{eq:Omega_rho_f_delta_definition_copy}
make the change of variables \eqref{eq:x_and_y_through_t_and_u}
from the Exercise \ref{exer:logarithmic_change_of_variables_in_the_distance_rho}:
\par\smallskip
\begin{equation} \label{eq:Omega_with_change_of_variables}
\Omega_{\rho,f}\,=\,\sup\,\Bigl\{\bigl|f(\hsp{3em})-f(\hsp{3em})\bigr|\colon\quad
u,v\in\ubrace{2em},\quad d(\,\ubrace{2em}\,,\,\ubrace{2em}\,)\le\ubrace{2em}\,\Bigr\}.
\end{equation}
\end{exercise}

\medskip

\begin{exercise}
The difference\quad$|f(\hsp{3em})-f(\hsp{3em})\bigr|$ 
in the right-hand side of \eqref{eq:Omega_with_change_of_variables}
can be written as
\par\smallskip
\[
|g(t)-g(u)|,\qquad\text{where}\qquad g(t)\eqdef\ubrace{4em}.
\]
\end{exercise}

\medskip

\begin{exercise}
Conclusion: $\Omega_{\rho,f}(\delta)$ can be written as $\Omega_{d,?}(?)$:
\par\smallskip
\[
\Omega_{\rho,f}(\delta)\,=\,\Omega_{d,\hsp{3em}}(\hsp{2em}).
\]
\par\bigskip\bigskip\noindent
In other words, the $\delta$-modulus of continuity
of a function $f\colon\PositiveRealNumbers\to\ComplexNumbers$\\[1ex]
with respect to the logarithmic distance $\rho$\\[1ex]
is equal to the $\ubrace{2em}$-modulus of continuity
of the function $\ubrace{4em}\colon\ubrace{2em}\to\ComplexNumbers$\\[1ex]
with respect to the standard distance $d$.
\end{exercise}

\clearpage
\section*{Some properties of $\boldsymbol{\cos}$}

\medskip\noindent
The functions $\cos$ and $\sin$ are ones of the simplest bounded functions
that do not have any limits at infinity.
Here we recall some well-known properties of the function $\cos$.

\bigskip

\begin{exercise} \label{exer:difference_of_cosines}
Recall the formula for the difference of cosines:
\par\smallskip
\[
\cos(t)-\cos(u)=\quad\quad\phantom{\sin}\quad\frac{t+u}{2}\quad\phantom{sin}\quad\frac{\hsp{3em}}{2}.
\]
\end{exercise}

\bigskip

\begin{exercise} \label{exer:standard_upper_bounds_for_sine}
Recall two standard upper bounds for $\sin(a)$, $a\in\RealNumbers$:
\par\smallskip
\[
\bigl|\sin(a)\bigr|\le\ubrace{3em},\qquad\qquad\bigl|\sin(a)\bigr|\le\ubrace{3em}.
\]
One of these upper bounds is a constant,
the other is not a constant and is more precise for small values of $a$.
\end{exercise}

\bigskip

\begin{exercise} \label{exer:cos_is_Lipschitz_continuous}
Using the results of the
Exercises \ref{exer:difference_of_cosines} and \ref{exer:standard_upper_bounds_for_sine}
prove that $\cos$ is Lipschitz-continuous:
\par\smallskip
\[
\bigl|\cos(t)-\cos(u)\bigr|\le \hsp{15em}
\]
\end{exercise}

\bigskip

\begin{exercise} \label{exer:max_of_cosine}
Recall in what points of the real line $\cos$ takes its maximum value:
\par\smallskip
\[
\cos(t)=1\qquad\Longleftrightarrow\qquad \hsp{15em}
\]
\end{exercise}

\bigskip

\begin{exercise} \label{exer:min_of_cosine}
Recall in what points of the real line $\cos$ takes its minimum value:
\par\smallskip
\[
\cos(u)=-1\qquad\Longleftrightarrow\qquad \hsp{15em}
\]
\end{exercise}

\clearpage
\section*{Example: $\boldsymbol{f(x)\eqdef\cos(\ln(x))}$}

\medskip
Consider the function $f\colon\PositiveRealNumbers\to\ComplexNumbers$ defined by
$f(x)\eqdef\cos(\ln(x))$.

\bigskip

\begin{exercise}
Find a good upper bound for the modulus of continuity of the function $f$
with respect to the logarithmic distance $\rho$:
\par\smallskip
\[
\Omega_{\rho,f}(\delta)\le\;?.
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\section*{Example: $\boldsymbol{f(x)\eqdef\cos\left(\sqrt{x}\right)}$}

\medskip\noindent
Consider the function $f\colon\PositiveRealNumbers\to\ComplexNumbers$ defined by
$f(x)\eqdef\cos\left(\sqrt{x}\right)$.

\bigskip

\begin{exercise}
Find a good lower bound for the modulus of continuity of the function $f$
with respect to the logarithmic distance $\rho$:
\par\smallskip
\[
\Omega_{\rho,f}(\delta)\ge\;?.
\]
\end{exercise}


\label{endlabel}
\end{document}
