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\begin{document}

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\section*{absolutely continuous function}

\hypertarget{absolutely_continuous_functions}{}\section*{{Absolutely continuous functions}}\label{absolutely_continuous_functions}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The basic idea behind a [[continuous function]] is that the output of the function can be made to change by only a small amount so long as the input is allowed to change by only a small amount. There are, of course, different ways to make this precise, including [[uniformly continuous functions]] and [[Lipschitz continuous functions]]. With an absolutely continuous function, you allow multiple changes to multiple inputs to be combined into a single total change (and you consider the [[absolute values]] of the changes, so that they won't cancel).

The result is a notion of function that gets along well with the [[fundamental theorem of calculus]] in the context of the [[Lebesgue integral]] on the [[real line]]. Absolute continuity is weaker than Lipschitz continuity but stronger than mere (pointwise) continuity.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

The first definition below is the most elementary; that the others are equivalent are important theorems.

Let $a$ and $b$ be [[real numbers]], and let $f$ be a real-valued [[function]] on the [[interval]] $[a,b]$. Then $f$ is \textbf{absolutely continuous} on $[a,b]$ iff:

\begin{defn}
\label{elementary}\hypertarget{elementary}{}
Given any [[positive number]] $\epsilon$, for some positive number $\delta$, given any [[natural number]] $n$ and any $2n$-tuple of elements of $[a,b]$, interpreted as an increasing $n$-tuple of nonoverlapping subintervals of $[a,b]$, if the total length of the intervals is less than $\delta$, then the total variation of $f$ on the intervals is less than $\epsilon$. That is (after $\epsilon$ and $\delta$), given $a \leq a_1 \leq b_1 \leq a_2 \leq b_2 \leq \cdots \leq a_n \leq b_n \leq b$, if

\begin{displaymath}
\sum_{i = 1}^n (b_i - a_i) \lt \delta ,
\end{displaymath}
then

\begin{displaymath}
\sum_{i = 1}^n {|{f(b_i) - f(a_i)}|} \lt \epsilon .
\end{displaymath}
\end{defn}
Various trivial variations of this may be met with: the comparison with $\delta$ and/or $\epsilon$ may be weak instead of strict; the number of subintervals may be infinite (so long as they are still nonoverlapping), since an infinite sum (of nonnegative numbers, as we have here) is simply a [[supremum]] of finite sums; and of course we may start by specifying that the $2n$ numbers come in order as the endpoints of the $n$ subintervals, rather than starting with any $2n$ numbers and then putting them in order and forming the subintervals from those. (Note that putting them in order is fine even in [[constructive analysis]], since choosing the $i$th element in order from a list of [[rational numbers]] is continuous, so may be extended constructively to real numbers, although we can't assume that the final list is a permutation of the original list.)

For the next definition, fix a model of [[nonstandard analysis]].

\begin{defn}
\label{nonstandard}\hypertarget{nonstandard}{}
Given any [[hypernatural number]] $n$ in the model and any $2n$-tuple of elements of the nonstandard extension of $[a,b]$, interpreted as an increasing $n$-tuple of nonoverlapping subintervals of $[a,b]$, if the total length of the intervals is [[infinitesimal number|infinitesimal]], then the total variation of $f$ on the intervals is infinitesimal. That is, given [[hyperreal numbers]] $a \leq a_1 \leq b_1 \leq a_2 \leq b_2 \leq \cdots \leq a_n \leq b_n \leq b$, if

\begin{displaymath}
\sum_{i = 1}^n (b_i - a_i) \approx 0 ,
\end{displaymath}
then

\begin{displaymath}
\sum_{i = 1}^n {|{f^*(b_i) - f^*(a_i)}|} \approx 0 .
\end{displaymath}
\end{defn}
See \hyperlink{Tuckey1993}{Tuckey 1993}, pages 34--36.

That the next definition is equivalent is the [[fundamental theorem of calculus]] for the [[Lebesgue integral]] on the [[real line]].

\begin{defn}
\label{Lebesgue}\hypertarget{Lebesgue}{}
There exists a Lebesgue-integrable function $g$ on $[a,b]$ such that $f(x) = f(a) + \int_{x=a} g(x) \,\mathrm{d}x$ for $x \in [a,b]$. (This is a [[semidefinite integral]].)

\end{defn}
In this case, $g$ must equal the [[derivative]] $f'$ [[almost everywhere]] on $[a,b]$. (So in particular, $f$ is differentiable almost everywhere with a Lebesgue-integrable derivative, although this is not enough without requiring that $f$ be an indefinite integral of its derivative.)

\begin{defn}
\label{Luzin}\hypertarget{Luzin}{}
The function $f$ is [[uniformly continuous]] on $[a,b]$, $f$ is of [[bounded variation]] on $[a,b]$, and the [[direct image]] under $f$ of any [[null subset]] of $[a,b]$ is null.

\end{defn}
\begin{defn}
\label{Stieltjes}\hypertarget{Stieltjes}{}
The [[Stieltjes measure]] $\mathrm{d}f$ is [[absolutely continuous measure|absolutely continuous]] with respect to [[Lebesgue measure]] on $[a,b]$.

\end{defn}
The last of these is the source of the term `absolutely continuous' as applied to measures.

\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

One may easily generalize the [[codomain]] of the elementary definition of absolutely continuous functions to any [[metric space]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The cube-root function is absolutely continuous (on any bounded interval) but not Lipschitz continuous (on any interval containing $0$).

The [[Cantor function]] is \emph{not} absolutely continuous, even though it is continuous, and differentiable almost everywhere, with a Lebesgue-integrable derivative.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Curtis Tuckey. 1993. Nonstandard Methods in the Calculus of Variations. CRC Press. \href{https://books.google.com/books?id=RvrC1mw9tIEC&q=%22absolutely+continuous%22#v=snippet&q=%22absolutely%20continuous%22&f=false}{Google books snippet}.

\end{itemize}
[[!redirects absolutely continuous function]] [[!redirects absolutely continuous functions]] [[!redirects absolutely continuous mapping]] [[!redirects absolutely continuous mappings]] [[!redirects absolutely continuous map]] [[!redirects absolutely continuous maps]]



\end{document}