Absolutely Continuous Functions Review

# Absolutely Continuous Functions Review

We will now review some of the recent material regarding absolutely continuous functions.

- On the
**Functions of Bounded Variation on Closed Intervals**page we said that if $f$ is a function defined on $[a, b]$ and $P = \{ a = x_0, x_1, ..., x_n = b \}$ is a partition of $[a, b]$ then the**Variation of $f$ Associated with $P$**is defined as:

\begin{align} \quad V(P, f) = \sum_{i=1}^{n} | f(x_i) - f(x_{i-1}) | \end{align}

- We said that $f$ is of
**Bounded Variation**on $[a, b]$ if there exists an $M \in \mathbb{R}$, $M > 0$ such that for all partitions $P$ on $[a, b]$ we have that $V(P, f) \leq M$.

- We noted a very important result - every function of bounded variation can be expressed as a difference of two increasing functions.

- On the
**Absolute Continuity**page we said that a function $f : [a, b] \to \mathbb{R}$ is**Absolutely Continuous**on $[a, b]$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $\{ (x_1, y_1), (x_2,y_2), ..., (x_n, y_n) \}$ is a finite collection of mutually disjoint open subintervals of $[a, b]$ such that $\displaystyle{\sum_{i=1}^{n} (y_i - x_i) < \delta}$ then:

\begin{align} \quad \sum_{i=1}^{n} | f(y_i) - f(x_i) | < \epsilon \end{align}

- We noted that every absolutely continuous function on $[a, b]$ is uniformly continuous on $[a, b]$ and hence continuous on $[a, b]$. Therefore absolute continuity is a stronger version of continuity.

- On the
**Absolutely Continuous Functions are of Bounded Variation**page we proved an important result. We proved that every absolutely continuous function is of bounded variation on $[a, b]$.

- On the
**Functions of Lebesgue Integrals**page we began to look at functions of Lebesgue integrals. In particular, if $f$ is Lebesgue integrable on $[a, b]$ we defined the new function $F : [a, b] \to \mathbb{R}$ by:

\begin{align} \quad F(x) = \int_a^x f(t) \: dt \end{align}

Property 1 | If $f$ is Lebesgue integrable on $[a, b]$ then $F$ is uniformly continuous on $[a, b]$. |
---|---|

Property 2 | If $f$ is Lebesgue integrable on $[a, b]$ then $F$ is of bounded variation on $[a, b]$. |

- On the
**Integral Criteria for Functions to be Zero Almost Everywhere**page we looked at two important results regarding when a function will be equal to zero almost everywhere in terms of integrals.

Lemma 1 | If $f$ is a nonnegative Lebesgue integrable function on $E$ and $\displaystyle{\int_E f = 0}$ then $f(x) = 0$ almost everywhere on $[a, b]$. |
---|---|

Lemma 2 | If $f$ is a Lebesgue measurable function and $F(x) = \int_a^x f(t) \: dt$ is such that $F(x) = 0$ for all $x \in [a, b]$ then $f(x) = 0$ almost everywhere on $[a, b]$. |

- On the
**The Derivative of Functions of Lebesgue Integrals**page we proved two results.

Theorem 1 | If $f$ is a bounded Lebesgue integrable function on $[a, b]$ and $\displaystyle{F(x) = \int_a^x f(t) \: dt}$ then $F'(x) = f(x)$ almost everywhere on $[a, b]$. |
---|---|

Theorem 2 | If $f$ is a Lebesgue integrable function on $[a, b]$ and $\displaystyle{F(x) = \int_a^x f(t) \: dt}$ then $F'(x) = f(x)$ almost everywhere on $[a, b]$. |

- Lastly, on the
**Classification of Absolutely Continuous Functions**page we proved that if $f$ is an absolutely continuous function on $[a, b]$ and $f'(x) = 0$ almost everywhere on $[a, b]$ then $f$ is a constant function on $[a, b]$.

- We used this to prove our major result which classified absolutely continuous functions. We proved that a real-valued function $F$ is absolutely continuous on $[a, b]$ if and only if there exists a Lebesgue integrable function $f$ on $[a, b]$ such that:

\begin{align} \quad F(x) = \int_a^x f(t) \: dt + F(a) \end{align}