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Section 6.3 Bounded Variation
Definition 6.33 . Bounded Variation.
Given \(f:[a,b]\rightarrow \mathbb{R}\) and a partition \(P\text{,}\) define
\begin{equation*}
V(f,P)=\sum_{i=1}^n\left|{f(x_i)-f(x_{i-1})}\right|.
\end{equation*}
Then define the variation of \(f\) over \([a,b]\) to be
\begin{equation*}
V_{[a,b]}(f)=\sup\{{V(f,P)}\,:\,{P\in\mathcal{P}[a,b]}\}\in[0,\infty].
\end{equation*}
We say \(f\) has bounded variation if \(V_{[a,b]}(f)<\infty\text{.}\)
Example 6.35 .
A function not of bounded variation: \(\begin{cases}
x\sin\left({\frac{\pi}{x}}\right) & x\neq 0\\
0 & x=0
\end{cases}\)
Theorem 6.37 . BV Functions Form Vector Space.
If \(f,g:[a,b]\rightarrow\mathbb{R}\) and \(\alpha\in\mathbb{R}\text{,}\) then
\begin{equation*}
\begin{aligned}
V_{[a,b]}(\alpha f)&=\left|{\alpha}\right|V_{[a,b]}(f);\\
V_{[a,b]}(f+g)&\leq V_{[a,b]}(f)+V_{[a,b]}(g).
\end{aligned}
\end{equation*}
In particular, \(\mathcal{BV}[a,b]\) is a vector space.
Proposition 6.38 . Lipschitz Implies BV.
Lipschitz functions have bounded variation.
Lemma 6.39 .
If \(f:[a,b]\rightarrow\mathbb{R}\) is differentiable, and \(f':[a,b]\rightarrow\mathbb{R}\) is bounded, then \(f\) is Lipschitz and in particular \(f\in\mathcal{BV}[a,b]\text{.}\)
Theorem 6.40 .
If \(f:[a,b]\rightarrow\mathbb{R}\) is differentiable and \(f':[a,b]\rightarrow\mathbb{R}\) is continuous, then \(f\in\mathcal{BV}[a,b]\text{.}\)
There are differentiable functions on closed intervals which are not of bounded variation.
Example 6.41 .
\begin{equation*}
\begin{aligned}
f:[0,1]&\rightarrow \mathbb{R}\\
f(x)&=
\begin{cases}
x^2\sin\left({\frac{1}{x^2}}\right) & x\neq 0\\
0 & x=0
\end{cases}.
\end{aligned}
\end{equation*}
Proposition 6.42 .
All monotone functions on a closed interval are of bounded variation.
We'll soon show that every function of bounded variation is a difference of two increasing functions. First, we need the following result.
Lemma 6.43 .
If \(f:[a,b]\rightarrow\mathbb{R}\) and \(c\in(a,b)\text{,}\) then \(V_{[a,b]}(f)=V_{[a,c]}(f)+V_{[c,b]}(f)\text{.}\)
Theorem 6.44 .
If \(f\in\mathcal{BV}[a,b]\text{,}\) then there are increasing functions \(g,h:[a,b]\rightarrow \mathbb{R}\) such that \(f=g-h\text{.}\)
Corollary 6.45 .
\(\mathcal{BV}[a,b]\subseteq \mathcal{R}[a,b]\text{.}\)
Theorem 6.46 .
Functions of bounded variation have only countably many discontinuities.
