FUNCTIONS OF BOUNDED VARIATION

This lecture introduces a class of functions that are differentiable almost everywhere.
For a function $f:[a,b]\rightarrow \mathbb{R}$, given a partition $P={a=x_0 less than x_1 less than \dots less than x_n=b}$, the quantity $t(P,f)$ is defined as $\sum_{i=1}^{n}|f(x_i)-f(x_{i-1})|$.
The total variation of $f$ is given by $T^b_a(f)=\sup_{P}t(P,f)$.
A function $f$ is said to be of if its total variation is finite, i.e., $T^b_a(f) less than \infty$.
If $f:[a,b]\rightarrow\mathbb{R}$ is Lipschitz continuous, meaning $|f(x) - f(y)| \le L|x - y|$, then $T^b_a(f) \le L(b-a) less than \infty$.
If $f:[a,b]\rightarrow\mathbb{R}$ is monotonic (increasing or decreasing), $T^b_a(f)=|f(b)-f(a)| less than \infty$.
An example of a continuous function that is of bounded variation is $f(x)=x^2\sin(1/x^2)$ for $x\in(0,1]$ and $f(0)=0$.
If $f$ is of bounded variation, then $|f|$ is also of bounded variation.
If $f$ and $g$ are BV, and $\alpha, \beta \in \mathbb{R}$, then the linear combination $\alpha f + \beta g$ is BV.
If $f$ and $g$ are BV, then their product $fg$ is BV.
For a real number $r$, the positive part is $r^+=\max(r,0)$ and the negative part is $r^-=-\min(r,0)$.
$r = r^+ - r^-$ and $|r| = r^+ + r^-$.
For a partition $P$, the positive variation $p(P,f) = \sum_{i=1}^{n}(f(x_i)-f(x_{i-1}))^+$ and the negative variation $n(P,f) = \sum_{i=1}^{n}(f(x_i)-f(x_{i-1}))^-$ are defined.
The total variation is $t(P,f) = p(P,f) + n(P,f)$.
Also, $f(b)-f(a) = p(P,f) - n(P,f)$.
The total positive variation $P^b_a(f) = \sup_P p(P,f)$ and total negative variation $N^b_a(f) = \sup_P n(P,f)$ are defined.
If $f$ is a function of BV, then $T^b_a(f)=P^b_a(f)+N^b_a(f)$ and $f(b)-f(a)=P^b_a(f)-N^b_a(f)$.

Видео FUNCTIONS OF BOUNDED VARIATION канала Dr P K Chaurasia